Chapter 5

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. $$f(x)=e^{-x^{2} / 2}$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=x^{3}+5 x-2$$

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. $$f(x)=x e^{x^{2}}$$

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

Use any method to find the relative extrema of the function \(f\). $$f(x)=x^{4}-2 x^{2}+7$$

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. $$f(x)=\ln \left(1+x^{2}\right)$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=x(x-1)^{2}$$

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. $$f(x)=x^{2} \ln x$$

Sketch the general shape of the graph of \(y=x^{1 / n},\) and then explain in words what happens to the shape of the graph as \(n\) increases if (a) \(n\) is a positive even integer (b) \(n\) is a positive odd integer.

Use any method to find the relative extrema of the function \(f\). $$f(x)=x^{4}+2 x^{3}$$

Analyze the trigonometric function \(f\) over the specified interval, stating where \(f\) is increasing. decreasing, concave up, and concave down, and stating the \(x\) coordinates of all inflection points. Confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=\cos x ;[0,2 \pi]$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=2 x^{2}-x^{4}$$

Analyze the trigonometric function \(f\) over the specified interval, stating where \(f\) is increasing. decreasing, concave up, and concave down, and stating the \(x\) coordinates of all inflection points. Confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=\sin ^{2} 2 x ;[0, \pi]$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=(2 x-1)^{5}$$

Analyze the trigonometric function \(f\) over the specified interval, stating where \(f\) is increasing. decreasing, concave up, and concave down, and stating the \(x\) coordinates of all inflection points. Confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=\tan x ;(-\pi / 2, \pi / 2)$$

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$2 x+3 x^{2 / 3}$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=x^{4 / 5}$$

Analyze the trigonometric function \(f\) over the specified interval, stating where \(f\) is increasing. decreasing, concave up, and concave down, and stating the \(x\) coordinates of all inflection points. Confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=2 x+\cot x ;(0, \pi)$$

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$4 x-3 x^{4 / 3}$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=2 x+x^{2 / 3}$$

Analyze the trigonometric function \(f\) over the specified interval, stating where \(f\) is increasing. decreasing, concave up, and concave down, and stating the \(x\) coordinates of all inflection points. Confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=\sin x \cos x ;[0, \pi]$$

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$x \sqrt{3-x}$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=\frac{x^{2}}{x^{2}+1}$$

Analyze the trigonometric function \(f\) over the specified interval, stating where \(f\) is increasing. decreasing, concave up, and concave down, and stating the \(x\) coordinates of all inflection points. Confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=\cos ^{2} x-2 \sin x ;[0,2 \pi]$$

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$4 x^{1 / 3}-x^{4 / 3}$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=\frac{x}{x+2}$$

In each part sketch a continuous curve \(y=f(x)\) with the stated properties. (a) \(f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)>0\) for all \(x\) (b) \(f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)<0\) for \(x<2, f^{\prime \prime}(x)>0\) for \(x>2\) (c) \(f(2)=4, f^{\prime \prime}(x)<0\) for \(x \neq 2\) and \(\lim _{x \rightarrow 2^{+}} f^{\prime}(x)=+\infty\) \(\lim _{x \rightarrow 2^{-}} f^{\prime}(x)=-\infty\)

Give a complete graph of the function, and identify the location of all critical points and inflection poir s. Check your work with a graphing utility. $$\frac{8(\sqrt{x}-1)}{x}$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=\ln \left(1+x^{2}\right)$$

In each part sketch a continuous curve \(y=f(x)\) with the stated properties. (a) \(f(2)=4, \quad f^{\prime}(2)=0, \quad f^{\prime \prime}(x)<0\) for all \(x\) (b) \(f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)>0\) for \(x<2, f^{\prime \prime}(x)<0\) for \(x>2\) (c) \(f(2)=4, \quad f^{\prime \prime}(x)>0\) for \(x \neq 2\) and \(\lim _{x \rightarrow 2^{+}} f^{\prime}(x)=-\infty$$\lim _{x \rightarrow 2^{-}} f^{\prime}(x)=+\infty\)

Use any method to find the relative extrema of the function \(f\). $$f(x)=x^{2} e^{x}$$

In each part, assume that \(a\) is a constant and find the inflection points, if any. (a) \(f(x)=(x-a)^{3}\) (b) \(f(x)=(x-a)^{4}\)

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x+\sin x$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=\left|x^{2}-4\right|$$

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x-\cos x$$

Use any method to find the relative extrema of the function \(f\). $$f(x)=\left\\{\begin{array}{ll} 9-x, & x \leq 3 \\\x^{2}-3, & x>3\end{array}\right.$$

If \(f\) is increasing on an interval \([0, b),\) then it follows from Definition 5.1.1 that \(f(0)0,\) and confirm the inequality with a graphing utility. [Hint: Show that the function \(\left.f(x)=1+\frac{1}{3} x-\sqrt[3]{1+x} \text { is increasing on }[0,+\infty) .\right]\)

Find the relative extrema in the interval \(0

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$\sin x+\cos x$$

If \(f\) is increasing on an interval \([0, b),\) then it follows from Definition 5.1.1 that \(f(0)

Find the relative extrema in the interval \(0

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$\sqrt{3} \cos x+\sin x$$

Find the relative extrema in the interval \(0

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$\sin ^{2} x, \quad 0 \leq x \leq 2 \pi$$

If \(f\) is increasing on an interval \([0, b),\) then it follows from Definition 5.1.1 that \(f(0)

Find the relative extrema in the interval \(0< x <2 \pi,\) and confirm that your results are consistent with the graph of \(f\) generated with a graphing utility. $$f(x)=\frac{\sin x}{2-\cos x}$$

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x \tan x, \quad-\pi / 2

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval; then use those graphs to estimate the \(x\) -coordinates of the inflection points of \(f\), the intervals on which \(f\) is concave up or down, and the intervals on which \(f\) is increasing or decreasing. Check your estimates by graphing \(f\). $$f(x)=x^{4}-24 x^{2}+12 x, \quad-5 \leq x \leq 5$$

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=x \ln x$$

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval; then use those graphs to estimate the \(x\) -coordinates of the inflection points of \(f\), the intervals on which \(f\) is concave up or down, and the intervals on which \(f\) is increasing or decreasing. Check your estimates by graphing \(f\). $$f(x)=\frac{1}{1+x^{2}}, \quad-5 \leq x \leq 5$$