Chapter Summary and Practice

Problem
A circle has radius 4 cm increasing at 0.5 cm/s. Find its area rate with respect to radius and its area rate with respect to time.

1. 1.Since $A=\pi r^2$, $dA/dr=2\pi r=8\pi$ cm² per cm at the given radius.
2. 2.The chain rule gives $dA/dt=(dA/dr)(dr/dt)=8\pi (0.5)=4\pi$ cm²/s.
3. 3.The numbers and units differ because the denominator variables differ. Over a short time $\Delta t$, the area increase is approximately $4\pi \Delta t$ cm² near this instant.

Problem
For $f(x)=x^3-3x$ on $[-2,3]$, find monotonicity intervals, local extrema, and absolute extrema.

1. 1.Differentiate: $f^{\prime }(x)=3(x-1)(x+1)$. Its signs are positive before $-1$, negative between $-1$ and 1, and positive after 1.
2. 2.Restricted to the domain, $f$ increases on $[-2,-1]$ and $[1,3]$, and decreases on $[-1,1]$.
3. 3.The local maximum is $f(-1)=2$, and the local minimum is $f(1)=-2$.
4. 4.For absolute extrema compare $f(-2)=-2$, $f(-1)=2$, $f(1)=-2$, and $f(3)=18$.
5. 5.The absolute maximum is 18 at 3. The absolute minimum is $-2$, attained at both $-2$ and 1. A local peak need not be the absolute maximum, and absolute extrema need not have unique locations.

Problem
Classify the critical point of $g(x)=x^4+2$.

1. 1.The derivative $g^{\prime }(x)=4x^3$ vanishes at 0, and $g^{\prime \prime }(0)=0$. The second derivative test is inconclusive.
2. 2.The first derivative is negative for $x<0$ and positive for $x>0$, giving a local minimum at 0.
3. 3.Directly, $x^4+2\ge 2$ for every real input, with equality only at 0. Thus this local minimum is also the absolute minimum, of value 2.
4. 4.There is no maximum on the real line because the output grows without bound. Neither conclusion followed from a zero second derivative alone.

Problem
Show that $f(x)=\log x/x$ has its greatest value at $x=e$, for $x>0$. Here $\log$ is the natural logarithm.

1. 1.The quotient rule gives $f^{\prime }(x)=(1-\log x)/x^2$. The denominator is positive on the entire domain.
2. 2.The numerator is positive for $0<x<e$, zero at $e$, and negative for $x>e$.
3. 3.Thus $f$ increases all the way to $e$ and decreases thereafter. Its greatest value is $f(e)=1/e$.
4. 4.This is a whole-domain argument on $(0,\infty )$, not an application of the closed-interval theorem.

Problem
A total length $k>0$ is divided into the perimeter of a square and the circumference of a circle. Show that their combined area is least when the square side equals twice the circle radius.

1. 1.Let the square side be $s$ and circle radius be $r$. The constraint is $4s+2\pi r=k$, so $s=(k-2\pi r)/4$, with $0<r<k/(2\pi )$ for two nondegenerate shapes.
2. 2.The combined area is $A(r)=(k-2\pi r{)}^2/16+\pi r^2$. Differentiate: $A^{\prime }(r)=-(\pi /4)(k-2\pi r)+2\pi r$.
3. 3.Set it to zero: $k=2\pi r+8r$. Substituting in the constraint gives $s=2r$.
4. 4.The second derivative is ${\pi }^2/2+2\pi >0$ throughout the interval, and the quadratic has a unique global minimum at this feasible stationary input.
5. 5.The actual dimensions are $r=k/(2\pi +8)$ and $s=2k/(2\pi +8)$. The result balances contributions under a common length constraint.

Problem
Find absolute extrema of $f(x)={\cos }^{2}x+\sin x$ on $[0,\pi ]$.

1. 1.Use ${\cos }^{2}x=1-{\sin }^{2}x$ to obtain $f(x)=1+\sin x-{\sin }^{2}x=5/4-(\sin x-1/2{)}^2$.
2. 2.On this interval $0\le \sin x\le 1$. The largest output is $5/4$, attained when $\sin x=1/2$, at $x=\pi /6$ and $5\pi /6$.
3. 3.For a derivative check, $f^{\prime }(x)=\cos x(1-2\sin x)$. The interior critical inputs are $\pi /6$, $\pi /2$, and $5\pi /6$; add 0 and $\pi$.
4. 4.At 0, $\pi /2$, and $\pi$, the value is 1. Thus the absolute minimum is 1 at all three inputs, and the maximum is $5/4$ at the other two.
5. 5.Keeping all ties matters. The second derivative test is unnecessary for the absolute comparison once the candidate values have been evaluated.