Chapter 5

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$x^{2}-2 x-3$$

In each part, sketch the graph of a function \(f\) with the stated properties, and discuss the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) (a) The function \(f\) is concave up and increasing on the interval \((-\infty,+\infty)\) (b) The function \(f\) is concave down and increasing on the interval \((-\infty,+\infty)\) (c) The function \(f\) is concave up and decreasing on the interval \((-\infty,+\infty)\) (d) The function \(f\) is concave down and decreasing on the interval \((-\infty,+\infty)\)

In each part, sketch the graph of a continuous function \(f\) with the stated properties. (a) \(f\) is concave up on the interval \((-\infty,+\infty)\) and has exactly one relative extremum. (b) \(f\) is concave up on the interval \((-\infty,+\infty)\) and has no relative extrema. (c) The function \(f\) has exactly two relative extrema on the interval \((-\infty,+\infty),\) and \(f(x) \rightarrow+\infty\) as \(x \rightarrow+\infty\) (d) The function \(f\) has exactly two relative extrema on the interval \((-\infty,+\infty),\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow+\infty\)

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$x^{3}-3 x+1$$

In each part, sketch the graph of a function \(f\) with the stated properties. (a) \(f\) is increasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave up on \((0,+\infty)\) (b) \(f\) is increasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave down on \((0,+\infty)\) (c) \(f\) is decreasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave up on \((0,+\infty)\) (d) \(f\) is decreasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave down on \((0,+\infty)\)

In each part, sketch the graph of a continuous function \(f\) with the stated properties. (a) \(f\) has exactly one relative extremum on \((-\infty,+\infty),\) and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\) (b) \(f\) has exactly two relative extrema on \((-\infty,+\infty),\) and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\) (c) \(f\) has exactly one inflection point and one relative extremum on \((-\infty,+\infty)\) (d) \(f\) has infinitely many relative extrema, and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\)

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$x^{3}-3 x+1$$

(a) Use both the first and second derivative tests to show that \(f(x)=3 x^{2}-6 x+1\) has a relative minimum at \(x=1\) (b) Use both the first and second derivative tests to show that \(f(x)=x^{3}-3 x+3\) has a relative minimum at \(x=1\) and a relative maximum at \(x=-1\)

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$2 x^{3}-3 x^{2}+12 x+9$$

(a) Use both the first and second derivative tests to show that \(f(x)=\sin ^{2} x\) has a relative minimum at \(x=0\) (b) Use both the first and second derivative tests to show that \(g(x)=\tan ^{2} x\) has a relative minimum at \(x=0\) (c) Give an informal verbal argument to explain without calculus why the functions in parts (a) and (b) have relative minima at \(x=0\).

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$x^{4}+2 x^{3}-1$$

(a) Show that both of the functions \(f(x)=(x-1)^{4}\) and \(g(x)=x^{3}-3 x^{2}+3 x-2\) have stationary points at \(x=1\) (b) What does the second derivative test tell you about the nature of these stationary points? (c) What does the first derivative test tell you about the nature of these stationary points?

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$x^{4}-2 x^{2}-12$$

(a) Show that \(f(x)=1-x^{5}\) and \(g(x)=3 x^{4}-8 x^{3}\) both have stationary points at \(x=0\) (b) What does the second derivative test tell you about the nature of these stationary points? (c) What does the first derivative test tell you about the nature of these stationary points?

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$3 x^{5}-5 x^{3}$$

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=x^{3}+3 x^{2}-9 x+1\) (b) \(f(x)=x^{4}-6 x^{2}-3\)

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$3 x^{4}+4 x^{3}$$

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=2 x^{3}-6 x+7\) (b) \(f(x)=3 x^{4}-4 x^{3}\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=x^{2}-5 x+6$$

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=\frac{x}{x^{2}+2}\) (b) \(f(x)=x^{2 / 3}\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=4-3 x-x^{2}$$

Give a complete graph of the polynomial, and label the coordinates of the stationary points and inflection points. Check your work with a graphing utility. $$x^{5}+5 x^{4}$$

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=\frac{x^{2}-3}{x^{2}+1}\) (b) \(f(x)=\sqrt[3]{x+2}\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=(x+2)^{3}$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{2 x}{x-3}$$

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=x^{1 / 3}(x+4)\) (b) \(f(x)=\cos 3 x\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=5+12 x-x^{3}$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{x}{x^{2}-1}$$

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=x^{4 / 3}-6 x^{1 / 3}\) (b) \(f(x)=|\sin x|\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=3 x^{4}-4 x^{3}$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{x^{2}}{x^{2}-1}$$

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=x^{4}-8 x^{2}+16$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{x^{2}-1}{x^{2}+1}$$

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=\frac{x^{2}}{x^{2}+2}$$

Use the given derivative to find the \(x\) coordinates of all critical points of \(f\), and determine whether a relative maximum, relative minimum, or neither occurs there. (a) \(f^{\prime}(x)=x^{3}\left(x^{2}-5\right)\) (b) \(f^{\prime}(x)=x e^{-x}\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=\frac{x}{x^{2}+2}$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{2 x^{2}-1}{x^{2}}$$

Use the given derivative to find the \(x\) coordinates of all critical points of \(f\), and determine whether a relative maximum, relative minimum, or neither occurs there. (a) \(f^{\prime}(x)=x^{2}(2 x+1)(x-1)\) (b) \(f^{\prime}(x)=\frac{9-4 x^{2}}{\sqrt[3]{x+1}}\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=\sqrt[3]{x+2}$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{x^{3}-1}{x^{3}+1}$$

Find the relative extrema using both the first and second derivative tests. $$f(x)=1-4 x-x^{2}$$

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=x^{2 / 3}$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{8}{4-x^{2}}$$

Find the relative extrema using both the first and second derivative tests. $$f(x)=2 x^{3}-9 x^{2}+12 x$$

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=x^{1 / 3}(x+4)$$

Give a complete graph of the rational function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label them with their equations. Check your work with a graphing utility. $$\frac{x-1}{x^{2}-4}$$

Find the relative extrema using both the first and second derivative tests. $$f(x)=\sin ^{2} x, \quad 0

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up. (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=x^{4 / 3}-x^{1 / 3}$$

The graph of the rational function crosses a horizontal asymptote. Give a complete graph of the function, and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes, and label the point(s) where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$\frac{3 x^{2}-4 x-4}{x^{2}}$$

Find the relative extrema using both the first and second derivative tests. $$f(x)=\frac{1}{2} x-\sin x, \quad 0