Chapter 4

Let \(f\) have a derivative on an interval \(I\). Prove that between successive distinct zeros of \(f^{\prime}\) there can be at most one zero of \(f .\) Hint: Try a proof by contradiction and use Rolle's Theorem (Problem 22).

A rectangle has two corners on the \(x\) -axis and the other two on the parabola \(y=12-x^{2}\), with \(y \geq 0\) (Figure 25). What are the dimensions of the rectangle of this type with maximum area?

The first derivative \(f^{\prime}\) is given. Find all values of \(x\) that make the function \(f(a)\) a local minimum and \((b)\) a local maximum. \(f^{\prime}(x)=(x-1)^{2}(x-2)^{2}(x-3)(x-4)\)

Linear approximations provide particularly good approximations near points of inflection. Using a graphing calculator, investigate this behavior in Problems 34-36. Graph \(y=\cos x\) and its linear approximation \(L(x)=-x+\pi / 2\) at \(x=\pi / 2 .\)

Prove that a quadratic function has no point of inflection.

Sketch the graph of a function with the given properties. \(f\) has domain \([0,6]\), but is not necessarily continuous, and \(f\) does not attain a maximum.

36\. Show that the differential equation $$ \frac{d y}{d t}=a y+b, y(0)=y_{0} $$ has solution $$ y=\left(y_{0}+\frac{b}{a}\right) e^{a t}-\frac{b}{a} $$ Assume that \(a \neq 0\).

The wolf population \(P\) in a certain state has been growing at a rate proportional to the cube root of the population size. The population was estimated at 1000 in 1980 and at 1700 in 1990 . (a) Write the differential equation for \(P\) at time \(t\) with the two corresponding conditions. (b) Solve the differential equation. (c) When will the wolf population reach \(4000 ?\)

A rectangle has two corners on the \(x\) -axis and the other two on the curve \(y=\cos x\), with \(-\pi / 2

. Let \(g\) be continuous on \([a, b]\) and suppose that \(g^{\prime \prime}(x)\) exists for all \(x\) in \((a, b)\). Prove that if there are three values of \(x\) in \([a, b]\) for which \(g(x)=0\) then there is at least one value of \(x\) in \((a, b)\) such that \(g^{\prime \prime}(x)=0\).

The first derivative \(f^{\prime}\) is given. Find all values of \(x\) that make the function \(f(a)\) a local minimum and \((b)\) a local maximum. \(f^{\prime}(x)=(x-1)^{2}(x-2)^{2}(x-3)^{2}(x-4)^{2}\)

Linear approximations provide particularly good approximations near points of inflection. Using a graphing calculator, investigate this behavior in Problems 34-36. Find the linear approximation to the curve \(y=\) \((x-1)^{5}+3\) at its point of inflection. Graph both the function and its linear approximation in the neighborhood of the inflection point.

Sketch the graph of a function with the given properties. \(f\) has domain \([0,6]\), but is not necessarily continuous, and \(f\) attains neither a maximum nor a minimum.

Of all right circular cylinders with a given surface area, find the one with the maximum volume. Note: The ends of the cylinders are closed.

Prove that, if \(f^{\prime}(x)\) exists and is continuous on an interval \(I\) and if \(f^{\prime}(x) \neq 0\) at all interior points of \(I\), then either \(\bar{f}\) is increasing throughout \(I\) or decreasing throughout \(I .\) Hint: Use the Intermediate Value Theorem to show that there cannot be two points \(x_{1}\) and \(x_{2}\) of \(I\) where \(f^{\prime}\) has opposite signs.

Prove that if \(\left|f^{\prime}(x)\right| \leq M\) for all \(x\) in \((a, b)\) and if \(x_{1}\) and \(x_{2}\) are any two points in \((a, b)\) then $$ \left|f\left(x_{2}\right)-f\left(x_{1}\right)\right| \leq M\left|x_{2}-x_{1}\right| $$

Find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\).

Suppose \(f^{\prime}(x)=(x-3)(x-2)^{2}(x-1)\) and \(f(2)=0\). Sketch a graph of \(y=f(x)\).

\mathrm{\\{} E X P L ~ B e s i d e s ~ p r o v i d i n g ~ a n ~ e a s y ~ w a y ~ t o ~ d i f f e r e n t i a t e ~ p r o d u c t s , ~ l o g . ~ arithmic differentiation also provides a measure of the relative or fractional rate of change, defined as \(y^{\prime} / y .\) We explore this concept in Problems 39-42. Show that the relative rate of change of \(e^{k t}\) as a function of \(t\) is \(k\)

An object thrown from the edge of a 42 -foot cliff follows the path given by \(y=-\frac{2 x^{2}}{25}+x+42\) (Figure 10\()\). An observer stands 3 feet from the bottom of the cliff. (a) Find the position of the object when it is closest to the observer. (b) Find the position of the object when it is farthest from the observer.

Of all rectangles with a given diagonal, find the one with the maximum area.

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) is differentiable, has domain \([0,6]\), and has two local maxima and two local minima on \((0,6)\).

Use the Monotonicity Theorem to prove each statement if \(0\frac{1}{y}\)

\mathrm{\\{} E X P L ~ B e s i d e s ~ p r o v i d i n g ~ a n ~ e a s y ~ w a y ~ t o ~ d i f f e r e n t i a t e ~ p r o d u c t s , ~ l o g . ~ arithmic differentiation also provides a measure of the relative or fractional rate of change, defined as $y^{\prime} / y . Show that the relative rate of change of any polynomial approaches zero as the independent variable approaches infinity.

A function \(f\) is said to be nondecreasing on an interval \(I\) if \(x_{1}

A humidifier uses a rotating disk of radius \(r\), which is partially submerged in water. The most evaporation occurs when the exposed wetted region (shown as the upper shaded region in Figure 27 ) is maximized. Show that this happens when \(h\) (the distance from the center to the water) is equal to \(r / \sqrt{1+\pi^{2}}\).

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) is differentiable, has domain \([0,6]\), and has three local maxima and two local minima on \((0,6)\).

Consider a general quadratic curve \(y=a x^{2}+b x+c\). Show that such a curve has no inflection points.

\mathrm{\\{} E X P L ~ B e s i d e s ~ p r o v i d i n g ~ a n ~ e a s y ~ w a y ~ t o ~ d i f f e r e n t i a t e ~ p r o d u c t s , ~ l o g . ~ arithmic differentiation also provides a measure of the relative or fractional rate of change, defined as $y^{\prime} / y . Prove that if the relative rate of change is a positive constant then the function must represent exponential growth.

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=3 x+1 $$

. Prove that, if \(f\) is continuous on \(I\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x) \geq 0\) on the interior of \(I\), then \(f\) is nondecreasing on I. Similarly, if \(f^{\prime}(x) \leq 0\), then \(f\) is nonincreasing on \(I\).

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), and has one local minimum and one local maximum on \((0,6)\).

Show that the curve \(y=a x^{3}+b x^{2}+c x+d\) where \(a \neq 0\), has exactly one inflection point.

\mathrm{\\{} E X P L ~ B e s i d e s ~ p r o v i d i n g ~ a n ~ e a s y ~ w a y ~ t o ~ d i f f e r e n t i a t e ~ p r o d u c t s , ~ l o g . ~ arithmic differentiation also provides a measure of the relative or fractional rate of change, defined as $y^{\prime} / y Prove that if the relative rate of change is a negative constant then the function must represent exponential decay.

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=-2 x+3 $$

Prove that if \(f(x) \geq 0\) and \(f^{\prime}(x) \geq 0\) on \(I\), then \(f^{2}\) is nondecreasing on \(I .\)

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), and has one local minimum and no local maximum on \((0,6)\)

Consider a general quartic curve \(y=a x^{4}+b x^{3}+\) \(c x^{2}+d x+e\), where \(a \neq 0\). What is the maximum number of inflection points that such a curve can have?

Suppose that the cubic function \(f(x)\) has three real zeros, \(r_{1}, r_{2}\), and \(r_{3}\). Show that its inflection point has \(x\) -coordinate \(\left(r_{1}+r_{2}+r_{3}\right) / 3 .\) Hint: \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right)\left(x-r_{3}\right)\)

Assume that (1) world population continues to grow exponentially with growth constant \(k=0.0132,(2)\) it takes \(\frac{1}{2}\) acre of land to supply food for one person, and (3) there are \(13,500,000\) square miles of arable land in the world. How long will it be before the world reaches the maximum population? Note: There were \(6.4\) billion people in 2004 and 1 square mile is 640 acres.

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=\sqrt{x} $$

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) has domain \([0,6]\), but is not necessarily continuous, and has three local maxima and no local minimum on \((0,6)\).

The graph of \(y=f(x)\) depends on a parameter c. Using a CAS, investigate how the extremum and inflection points depend on the value of \(c .\) Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=x^{2} \sqrt{x}^{2}-c^{2} $$

Suppose that \(f^{\prime}(x)>0\) and \(g^{\prime}(x)>0\) for all \(x .\) What simple additional conditions (if any) are needed to guarantee that: (a) \(f(x)+g(x)\) is increasing for all \(x\); (b) \(f(x)=g(x)\) is increasing for all \(x\); (c) \(f(g(x))\) is increasing for all \(x\) ?

GG 44. The Census Bureau estimates that the growth rate \(k\) of the world population will decrease by roughly \(0.0002\) per year for the next few decades. In \(2004, k\) was \(0.0132\). (a) Express \(k\) as a function of time \(t\), where \(t\) is measured in years since 2004 . (b) Find a differential equation that models the population \(y\) for this problem. (c) Solve the differential equation with the additional condition that the population in \(2004(t=0)\) was \(6.4\) billion. (d) Graph the population \(y\) for the next 300 years. (e) With this model, when will the population reach a maximum? When will the population drop below the 2004 level?

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=x^{4 / 3} $$

Use the Mean Value Theorem to prove that $$ \lim _{x \rightarrow \infty}(\sqrt{x+2}-\sqrt{x})=0 $$

I have enough pure silver to coat 1 square meter of surface area. I plan to coat a sphere and a cube. What dimensions should they be if the total volume of the silvered solids is to be a maximum? A minimum? (Allow the possibility of all the silver going onto one solid.)

Sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. \(f\) has domain \([0,6]\), but is not necessarily continuous, and has two local maxima and no local minimum on \((0,6)\).

The graph of \(y=f(x)\) depends on a parameter c. Using a CAS, investigate how the extremum and inflection points depend on the value of \(c .\) Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=\frac{c x}{4+(c x)^{2}} $$