AP Calculus AB Unit 5: Analytical Applications of Differentiation

The Mean Value Theorem and Extreme Value Theorem, critical points, increasing and decreasing intervals, the first and second derivative tests, the candidates test on a closed interval, concavity and points of inflection, connecting $f$, $f'$ and $f''$, optimization, and the behavior of implicitly defined curves.

CED topics in this unit: 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10, 5.11, 5.12.

Key formulas and facts

• MVT: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f'(c)=\frac{f(b)-f(a)}{b-a}$ for some $c$ in $(a,b)$
• EVT: if $f$ is continuous on $[a,b]$, then $f$ has an absolute maximum and an absolute minimum on $[a,b]$
• Critical point: $f'(c)=0$ or $f'(c)$ does not exist (with $c$ in the domain of $f$)
• First derivative test: $f'$ changes from $+$ to $-$ at a relative maximum, from $-$ to $+$ at a relative minimum
• Second derivative test: $f'(c)=0$ and $f''(c)<0$ gives a relative maximum; $f''(c)>0$ gives a relative minimum
• Point of inflection: $f''$ changes sign (equivalently, $f'$ changes from increasing to decreasing or the reverse)
• Candidates test: compare $f$ at the critical points and at the endpoints

Worked examples

• MVT: $f(x)=x^2-4x$ on $[0,5]$: $\frac{f(5)-f(0)}{5}=1$, and $2c-4=1$ gives $c=\frac52$
• Candidates test: $f(x)=x^3-3x$ on $[0,2]$: $f(0)=0$, $f(1)=-2$, $f(2)=2$; the absolute maximum is $2$ and the absolute minimum is $-2$
• Inflection: $f(x)=x^4-4x^3$ has $f''(x)=12x(x-2)$, which changes sign at $x=0$ and at $x=2$
• From $f'$: if $f'(x)=(x-1)^2(x+2)$, then $f'$ changes sign only at $x=-2$ (from $-$ to $+$), so $f$ has a relative minimum at $x=-2$ and no extremum at $x=1$
• Optimization: a rectangle with perimeter $40$ has area $A=x(20-x)$; $A'=20-2x=0$ at $x=10$ and $A''=-2<0$, so the largest area is $100$

Common traps

• Using the MVT without checking continuity on $[a,b]$ and differentiability on $(a,b)$.
• Calling a point with $f''(c)=0$ an inflection point without a sign change of $f''$ (for example $x^4$ at $0$).
• Leaving out the endpoints in an absolute-extremum question.
• Justifying with 'the graph goes up then down': state the sign change of $f'$.
• Reading a graph of $f'$ as if it were the graph of $f$.

Practice questions

78 multiple-choice and 5 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

$f(x)=\sqrt{x}$ satisfies the hypotheses of the Mean Value Theorem on $[1,9]$. What value of $c$ does the theorem guarantee?

1. $5$
2. $2$
3. $4$
4. $3$

Question 2 · Multiple choice

The function $f$ is given by $f(x)=x^4-18x^2$. Over which intervals is $f$ decreasing?

1. $(-\infty,-3)$ and $(0,3)$
2. $(-3,0)$ and $(3,\infty)$
3. $(-3,3)$
4. $(-\infty,0)$

Question 3 · Multiple choice

The derivative of $f$ is $f'(x)=(x-1)^2(x-4)e^{-x}$. Which statement about the relative extrema of $f$ is true?

1. $f$ has a relative maximum at $x=1$ and a relative minimum at $x=4$
2. $f$ has relative minima at $x=1$ and at $x=4$
3. $f$ has a relative minimum at $x=4$ only
4. $f$ has a relative maximum at $x=4$ only

Question 4 · Multiple choice

What is the absolute maximum value of $f(x)=x^3-12x^2+45x-10$ on the closed interval $[0,7]$?

1. $44$
2. $60$
3. $40$
4. $-10$

Question 5 · Multiple choice

On which interval is the graph of $f(x)=xe^{-x}$ concave up?

1. $(1,\infty)$
2. $(-\infty,2)$
3. $(-\infty,1)$
4. $(2,\infty)$

Question 6 · Multiple choice

The critical points of $f(x)=x^3-27x+4$ are $x=\pm3$. What is $f''(-3)$, and what does the second derivative test say about $f$ at $x=-3$?

1. $f''(-3)=-18$; $f$ has a relative minimum at $x=-3$
2. $f''(-3)=-18$; $f$ has a relative maximum at $x=-3$
3. $f''(-3)=18$; $f$ has a relative minimum at $x=-3$
4. $f''(-3)=0$; the test gives no conclusion

Question 7 · Multiple choice

The derivative of a function $g$ is $g'(x)=x^3-3x^2$. How many points of inflection does the graph of $g$ have?

1. $1$
2. $3$
3. $2$
4. $0$

Question 8 · Multiple choice

At which points on the curve $x^2+xy+y^2=12$ is the tangent line horizontal?

1. $(2,-4)$ and $(-2,4)$
2. $(4,-2)$ and $(-4,2)$
3. $(2,2)$ and $(-2,-2)$
4. $(0,2\sqrt3)$ and $(0,-2\sqrt3)$

Question 9 · Free response (9 points)

The function $g$ is twice differentiable for all real numbers. The table gives values of $g$ and $g'$ at selected values of $x$.

1. (a) Must there be a value $c$ in $(2,5)$ for which $g'(c)=0$? Justify your answer. [2 points]
2. (b) Must there be a value $c$ in $(5,7)$ for which $g(c)=5$? Justify your answer. [2 points]
3. (c) Must there be a value $c$ in $(5,10)$ for which $g''(c)=-0.8$? Justify your answer. [2 points]
4. (d) Use the data in the table to estimate $g'(6)$. [1 point]
5. (e) Must there be a value $c$ in $(0,2)$ for which $g'(c)=-1.5$? Justify your answer. [2 points]

Question 10 · Free response (9 points)

A rectangle sits on the $x$-axis, symmetric about the $y$-axis, with its two upper vertices on the graph of $y=27-x^2$. Its upper right vertex is $(x,27-x^2)$ with $x>0$.

1. (a) Write an expression for the area $A$ of the rectangle in terms of $x$, and give the domain of $A$. [2 points]
2. (b) Find the value of $x$ that gives the largest area. Justify your answer. [3 points]
3. (c) Find the largest area. [1 point]
4. (d) Find the value of $x$ that gives the largest perimeter, and find that perimeter. Justify that it is a maximum. [3 points]

Question 11 · Multiple choice

Let $f(x)=x^3-3x$. What value of $c$ in the open interval $(0,3)$ satisfies the conclusion of the Mean Value Theorem for $f$ on $[0,3]$?

1. $1$
2. $\sqrt{3}$
3. $\frac{3}{2}$
4. $\sqrt{2}$

Question 12 · Multiple choice

Let $f(x)=\ln x$. What value of $c$ satisfies the conclusion of the Mean Value Theorem for $f$ on $[1,e]$?

1. $\frac{1}{-1 + e}$
2. $-1 + e$
3. $\frac{1}{2} + \frac{e}{2}$
4. $e^{\frac{1}{2}}$

Question 13 · Multiple choice

A function $f$ is continuous on $[1,5]$ and differentiable on $(1,5)$, with $f(1)=4$ and $f(5)=12$. For which value of $k$ does the Mean Value Theorem guarantee a $c$ in $(1,5)$ with $f'(c)=k$?

1. $8$
2. $2$
3. $3$
4. $4$

Question 14 · Multiple choice

For which of the following functions does the Mean Value Theorem apply on the interval $[-1,1]$?

1. $f(x)=|x|$
2. $f(x)=x^{2/3}$
3. $f(x)=\dfrac{1}{x}$
4. $f(x)=x^3+x$

Question 15 · Multiple choice

Let $f(x)=e^x$. What value of $c$ satisfies the conclusion of the Mean Value Theorem for $f$ on $[0,2]$?

1. $1.000$
2. $1.161$
3. $3.195$
4. $1.097$

Question 16 · Multiple choice

On the closed interval $[-1,4]$, what is the absolute maximum value of $g(x)=2x^3-6x^2+1$?

1. $33$
2. $1$
3. $-7$
4. $9$

Question 17 · Multiple choice

Find all critical points of $g(x)=x^{2/3}(x-10)$.

1. $x=4$ only
2. $x=0$ and $x=10$
3. $x=0$ and $x=4$
4. $x=4$ and $x=10$

Question 18 · Multiple choice

Which function is guaranteed by the Extreme Value Theorem to have an absolute maximum on the given interval?

1. $f(x)=x^2$ on $(0,2)$
2. $f(x)=\dfrac{1}{x-1}$ on $[0,2]$
3. $f(x)=\sqrt x$ on $[0,4]$
4. $f(x)=\tan x$ on $[0,\pi]$

Question 19 · Multiple choice

For $1\le x\le4$, let $h(x)=2x+\dfrac{8}{x}$. What is the least value of $h$ on this interval?

1. $10$
2. $8$
3. $2$
4. $\frac{17}{2}$

Question 20 · Multiple choice

What is the absolute maximum value of $f(x)=xe^{-x}$ on $[0,3]$?

1. $e^{-1}$
2. $\frac{3}{e^{3}}$
3. $0$
4. $1$

Question 21 · Multiple choice

What is the absolute maximum value of $f(x)=\sin x+\cos x$ on $[0,\pi]$?

1. $1$
2. $\sqrt{2}$
3. $2$
4. $\frac{\sqrt{2}}{2}$

Question 22 · Multiple choice

How many critical points does $f(x)=x^4-4x^3$ have?

1. $1$
2. $2$
3. $3$
4. $4$

Question 23 · Multiple choice

What is the absolute minimum value of $f(x)=x^2-4\ln(x+1)$ on $[0,3]$?

1. $0$
2. $1 - 4 \ln{\left(2 \right)}$
3. $9 - 8 \ln{\left(2 \right)}$
4. $-1$

Question 24 · Multiple choice

On which interval is $f(x)=x^3-12x$ decreasing?

1. $(-2,2)$
2. $(2,\infty)$
3. $(-\infty,-2)$
4. $(0,2)$

Question 25 · Multiple choice

On which interval is $f(x)=x^2e^x$ decreasing?

1. $(0,\infty)$
2. $(-\infty,-2)$
3. $(-1,0)$
4. $(-2,0)$

Question 26 · Multiple choice

The derivative of $f$ is $f'(x)=(x-1)(x+2)^2$. On which interval is $f$ increasing?

1. $(-2,1)$
2. $(-\infty,-2)\cup(1,\infty)$
3. $(-\infty,1)$
4. $(1,\infty)$

Question 27 · Multiple choice

On which interval is $f(x)=\ln(x^2+1)$ increasing?

1. $(0,\infty)$
2. $(-\infty,0)$
3. $(-\infty,\infty)$
4. $(1,\infty)$

Question 28 · Multiple choice

On which interval in $(0,2\pi)$ is $f(x)=x-2\sin x$ increasing?

1. $\left(0,\frac{\pi}{3}\right)$
2. $\left(\frac{\pi}{3},\frac{5\pi}{3}\right)$
3. $\left(\frac{\pi}{6},\frac{11\pi}{6}\right)$
4. $(0,\pi)$

Question 29 · Multiple choice

The derivative of $f$ is $f'(x)=\dfrac{x-3}{(x+1)^2}$ for $x\neq-1$. On which interval is $f$ increasing?

1. $(-1,3)$
2. $(3,\infty)$
3. $(-\infty,-1)\cup(3,\infty)$
4. $(-\infty,3)$

Question 30 · Multiple choice

Where is the function $p(x)=x^4-18x^2$ increasing?

1. $(-\infty,-3)\cup(0,3)$
2. $(3,\infty)$
3. $(-3,0)\cup(3,\infty)$
4. $(-3,3)$

Question 31 · Multiple choice

The derivative of $f$ is $f'(x)=x(x-2)^2(x+1)$. At which value of $x$ does $f$ have a relative minimum?

1. $-1$
2. $2$
3. $1$
4. $0$

Question 32 · Multiple choice

Find the relative maximum value of $k(x)=2x^3+3x^2-72x$.

1. $208$
2. $-135$
3. $-4$
4. $3$

Question 33 · Multiple choice

Let $f(x)=x^{1/3}(x+4)$. At which value of $x$ does $f$ have a relative minimum?

1. $0$
2. $-4$
3. $1$
4. $-1$

Question 34 · Multiple choice

The derivative of $f$ is $f'(x)=e^x(x^2-4)$. At which value of $x$ does $f$ have a relative maximum?

1. $2$
2. $0$
3. $-2$
4. $-4$

Question 35 · Multiple choice

What is the relative minimum value of $f(x)=x\ln x$ for $x>0$?

1. $e^{-1}$
2. $0$
3. $-1$
4. $- \frac{1}{e}$

Question 36 · Multiple choice

What is the absolute maximum value of $f(x)=2x^3-3x^2-12x$ on $[-2,3]$?

1. $7$
2. $-4$
3. $-9$
4. $20$

Question 37 · Multiple choice

What is the absolute maximum value of $g(x)=x-2\sqrt{x}$ on $[0,9]$?

1. $0$
2. $-1$
3. $3$
4. $9$

Question 38 · Multiple choice

What is the absolute maximum value of $h(x)=\dfrac{x^2}{x^2+1}$ on $[-1,2]$?

1. $\frac{1}{2}$
2. $\frac{4}{5}$
3. $1$
4. $0$

Question 39 · Multiple choice

What is the absolute minimum value of $f(x)=\cos(2x)+2\cos x$ on $[0,\pi]$?

1. $- \frac{3}{2}$
2. $-1$
3. $-3$
4. $3$

Question 40 · Multiple choice

On which interval is the graph of $f(x)=x^3-6x^2$ concave down?

1. $(2,\infty)$
2. $(-\infty,2)$
3. $(0,4)$
4. $(-\infty,0)$

Question 41 · Multiple choice

At which values of $x$ does the graph of $f(x)=x^4-6x^2$ have points of inflection?

1. $x=-1$ and $x=1$
2. $x=0$ only
3. $x=-\sqrt3$ and $x=\sqrt3$
4. $x=-1$, $x=0$ and $x=1$

Question 42 · Multiple choice

On which interval is the graph of $f(x)=xe^x$ concave up?

1. $(-1,\infty)$
2. $(-2,\infty)$
3. $(-\infty,-2)$
4. $(0,\infty)$

Question 43 · Multiple choice

Let $f(x)=x^3-3x$. What does the second derivative test say about the critical point $x=1$?

1. relative maximum
2. relative minimum
3. neither: the test is inconclusive
4. point of inflection

Question 44 · Multiple choice

The derivative of $f$ is $f'(x)=x^2-4$. Using the second derivative test, at which value of $x$ does $f$ have a relative maximum?

1. $-2$
2. $2$
3. $0$
4. $4$

Question 45 · Multiple choice

Let $f(x)=x^4$. What does the second derivative test say about the critical point $x=0$?

1. relative minimum
2. neither: the test is inconclusive
3. relative maximum
4. point of inflection

Question 46 · Multiple choice

Let $g(x)=\sin x+\cos x$ on $(0,2\pi)$. At which value of $x$ does the second derivative test show a relative maximum?

1. $\frac{5 \pi}{4}$
2. $\frac{3 \pi}{4}$
3. $\frac{\pi}{2}$
4. $\frac{\pi}{4}$

Question 47 · Multiple choice

Let $f(x)=x-2\ln x$ for $x>0$. What is the relative minimum value of $f$?

1. $2$
2. $1$
3. $2 - 2 \ln{\left(2 \right)}$
4. $2 \ln{\left(2 \right)} + 2$

Question 48 · Multiple choice

Let $f(x)=x^2e^{-x}$. What is the relative maximum value of $f$?

1. $0$
2. $e^{-1}$
3. $\frac{4}{e^{2}}$
4. $\frac{2}{e^{2}}$

Question 49 · Multiple choice

The graph of $f'$, the derivative of $f$, is shown for $-3<x<4$. At which value of $x$ does $f$ have a relative maximum?

1. $2$
2. $0$
3. $3$
4. $-2$

Question 50 · Multiple choice

The graph of $f'$, the derivative of $f$, is shown for $-3<x<4$. At which value of $x$ does $f$ have a relative minimum?

1. $-2$
2. $0$
3. $2$
4. $4$

Question 51 · Multiple choice

The graph of $f'$, the derivative of $f$, is shown for $-3<x<4$. On which interval is the graph of $f$ concave up?

1. $(-2,2)$
2. $(0,3)$
3. $(2,4)$
4. $(-3,0)$

Question 52 · Multiple choice

The graph of $f'$, the derivative of $f$, is shown for $-3<x<4$. How many points of inflection does the graph of $f$ have on $(-3,4)$?

1. $2$
2. $1$
3. $3$
4. $0$

Question 53 · Multiple choice

The graph of $f'$, the derivative of $f$, is shown for $-3<x<4$. On which interval is $f$ both decreasing and concave down?

1. $(-2,0)$
2. $(0,2)$
3. $(-3,-2)$
4. $(3,4)$

Question 54 · Multiple choice

Using the graph of $f$ shown, at which value of $x$ is $f$ decreasing while its graph is concave up?

1. $- \frac{3}{2}$
2. $- \frac{1}{2}$
3. $\frac{1}{2}$
4. $\frac{3}{2}$

Question 55 · Multiple choice

Using the graph of $f$ shown, at which value of $x$ is $f$ increasing while its graph is concave down?

1. $\frac{1}{2}$
2. $- \frac{3}{2}$
3. $- \frac{1}{2}$
4. $\frac{3}{2}$

Question 56 · Multiple choice

The graph of $f''$, the second derivative of a function $f$, is shown for $-3<x<4$. How many points of inflection does the graph of $f$ have on this interval?

1. $3$
2. $2$
3. $4$
4. $1$

Question 57 · Multiple choice

The twice-differentiable function $f$ has the values in the table. Which statement about $f$ at $x=2$ is true?

1. f has a relative minimum at x = 2
2. f has a relative maximum at x = 2
3. f has a point of inflection at x = 2
4. f is increasing at x = 2

Question 58 · Multiple choice

Let $f(x)=\sqrt{x}$ on the interval $1<x<4$. Which describes the graph of $f$ on this interval?

1. increasing and concave down
2. increasing and concave up
3. decreasing and concave down
4. decreasing and concave up

Question 59 · Multiple choice

The second derivative of $f$ is $f''(x)=(x-1)(x-3)^2$. At which values of $x$ does the graph of $f$ have a point of inflection?

1. $x=1$ and $x=3$
2. $x=3$ only
3. $x=1$ only
4. There are none.

Question 60 · Multiple choice

For constants $k$ and $m$, let $f(x)=2x^3+kx^2+mx$. The graph of $f$ has its point of inflection at $x=0$, and $f$ has a relative minimum at $x=1$. What is $m$?

1. $-6$
2. $6$
3. $0$
4. $-2$

Question 61 · Multiple choice

A twice-differentiable function $f$ has $f(2)=3$, $f'(2)=0$ and $f''(2)=-1$. Which statement is true?

1. f has a relative minimum value of 3 at x = 2
2. f has a point of inflection at (2, 3)
3. f has a relative maximum value of 3 at x = 2
4. f has a relative maximum value of -1 at x = 2

Question 62 · Multiple choice

The derivative $f'$ is continuous on $[0,4]$, with values in the table. What is the least number of critical points that $f$ must have in the open interval $(0,4)$?

1. $1$
2. $3$
3. $0$
4. $2$

Question 63 · Multiple choice

Positive real numbers $x$ and $y$ satisfy $x+y=28$. Find the maximum value of $xy$.

1. $195$
2. $196$
3. $192$
4. $392$

Question 64 · Multiple choice

A farmer has $120$ feet of fencing to enclose a rectangular field along a straight river; no fence is needed along the river. What is the largest area that can be enclosed, in square feet?

1. $900$
2. $3600$
3. $1800$
4. $1600$

Question 65 · Multiple choice

What is the minimum value of $x+\dfrac{9}{x}$ for $x>0$?

1. $3$
2. $10$
3. $6$
4. $9$

Question 66 · Multiple choice

What is the $x$-coordinate of the point on the graph of $y=\sqrt x$ that is closest to the point $(3,0)$?

1. $3$
2. $2$
3. $\frac{5}{2}$
4. $\frac{1}{2}$

Question 67 · Multiple choice

An open-top box with a square base must hold $32$ cubic inches. What is the least possible surface area of the box, in square inches?

1. $32$
2. $48$
3. $64$
4. $40$

Question 68 · Multiple choice

A company's monthly revenue is $R(p)=p(200-4p)$ dollars when the price is $p$ dollars. What price maximizes the revenue?

1. $50$
2. $100$
3. $2500$
4. $25$

Question 69 · Multiple choice

A closed cylindrical can must have volume $16\pi$ cubic inches. What radius, in inches, minimizes the total surface area?

1. $4$
2. $1$
3. $2$
4. $\sqrt[3]{2}$

Question 70 · Multiple choice

A rectangle has its base on the $x$-axis and its two upper corners on the semicircle $y=\sqrt{25-x^2}$. What is the largest possible area of the rectangle?

1. $25$
2. $50$
3. $\frac{25}{2}$
4. $10$

Question 71 · Multiple choice

What is the maximum value of $f(x)=x\sin x$ on $[0,\pi]$?

1. $1.820$
2. $2.029$
3. $1.571$
4. $0.841$

Question 72 · Multiple choice

What is the shortest distance from the point $(0,4)$ to the parabola $y=x^2$?

1. $\frac{\sqrt{15}}{2}$
2. $4$
3. $\frac{\sqrt{14}}{2}$
4. $\frac{15}{4}$

Question 73 · Multiple choice

A cable runs from a station on one bank of a straight river $1$ mile wide to a point on the other bank $4$ miles downstream. Laying cable costs $5$ thousand dollars per mile under water and $3$ thousand dollars per mile on land. The cable crosses the river diagonally to a point $x$ miles downstream, then runs along the bank. What is the least total cost, in thousands of dollars?

1. $17$
2. $20$
3. $16$
4. $12$

Question 74 · Multiple choice

Equal squares of side $x$ are cut from the corners of a $12$-inch by $12$-inch sheet, and the sides are folded up to make an open box. What is the largest possible volume, in cubic inches?

1. $2$
2. $144$
3. $108$
4. $128$

Question 75 · Multiple choice

At which points on the curve $x^2+y^2-4x=0$ is the tangent line vertical?

1. $(2,2)$ and $(2,-2)$
2. $(2,0)$ only
3. $(0,0)$ only
4. $(0,0)$ and $(4,0)$

Question 76 · Multiple choice

How many points on the curve $y^2=x^3-3x+3$ have a horizontal tangent line?

1. $2$
2. $4$
3. $1$
4. $0$

Question 77 · Multiple choice

On the circle $x^2+y^2=100$, consider the point $(6,-8)$. Which statement is true there?

1. y is decreasing in x and the curve is concave up
2. y is increasing in x and the curve is concave down
3. y is decreasing in x and the curve is concave down
4. y is increasing in x and the curve is concave up

Question 78 · Multiple choice

The point $(1,2)$ is on the curve $4x^2+y^2=8$. What is the value of $\dfrac{d^2y}{dx^2}$ at this point?

1. $-2$
2. $4$
3. $-8$
4. $-4$

Question 79 · Multiple choice

For the curve $x^3+y^3=6xy$, which statement is true at the point $(3,3)$?

1. y is decreasing in x, with slope -1
2. y is increasing in x, with slope 1
3. the tangent line is horizontal
4. the tangent line is vertical

Question 80 · Multiple choice

What is the greatest $y$-coordinate of any point on the curve $x^2-xy+y^2=12$?

1. $4$
2. $2 \sqrt{3}$
3. $6$
4. $2$

Question 81 · Free response (9 points)

The function $f$ is twice differentiable on $(-4,3)$ except possibly where noted. The graph of its derivative $f'$, made of two line segments and a semicircle, is shown.

1. (a) Find the $x$-coordinate of each relative extremum of $f$ on $(-4,3)$. Classify each one and justify your answer. [2 points]
2. (b) On what open interval is the graph of $f$ concave down? Give a reason. [2 points]
3. (c) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason. [2 points]
4. (d) Find $f''(-3)$ and $f''(1)$. [3 points]

Question 82 · Free response (9 points)

A rectangle has its lower side on the $x$-axis and its two upper vertices on the graph of $y=12-x^2$, at $(x,12-x^2)$ and $(-x,12-x^2)$ with $0\le x\le\sqrt{12}$.

1. (a) Write an expression for the area $A(x)$ of the rectangle. [2 points]
2. (b) Find the value of $x$ that maximizes the area. Justify your answer. [3 points]
3. (c) What is the maximum area? Use the candidates test to confirm it. [2 points]
4. (d) Suppose the rectangle may be at most $3$ units wide. What is the largest possible area now? [2 points]

Question 83 · Free response (9 points)

A closed curve in the plane is given by $2x^2+2xy+y^2=18$.

1. (a) Show that $\dfrac{dy}{dx}=-\dfrac{2x+y}{x+y}$. [2 points]
2. (b) Find all points on the curve at which the tangent line is horizontal. [3 points]
3. (c) Find $\dfrac{d^2y}{dx^2}$ at $(-3,6)$. Does $y$, as a function of $x$ near this point, have a relative maximum or a relative minimum there? Justify your answer. [2 points]
4. (d) Find all points on the curve at which the tangent line is vertical. [2 points]