91Ó°ÊÓ

a. Find the critical points of the following functions on the domain or on the given interval. b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither. $$f(x)=\frac{4 x^{5}}{5}-3 x^{3}+5 \text { on }[-2,2]$$

Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. \(1 / \sqrt{119}\)

Determine the following indefinite integrals. Check your work by differentiation. $$\int 5 m\left(12 m^{3}-10 m\right) d m$$

An important question about many functions concerns the existence and location of fixed points. A fixed point of \(f\) is a value of \(x\) that satisfies the equation \(f(x)=x ;\) it corresponds to a point at which the graph off intersects the line \(y=x\). Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=5-x^{2}$$

Find all functions \(f\) whose derivative is \(f^{\prime}(x)=x+1\).

Find the intervals on which \(f\) is increasing and decreasing. $$f(x)=\cos ^{2} x \text { on }[-\pi, \pi]$$

Find the intervals on which \(f\) is increasing and decreasing. $$f(x)=x^{2 / 3}\left(x^{2}-4\right)$$

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(3 x^{1 / 3}+4 x^{-1 / 3}+6\right) d x$$

Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. \(1 / \sqrt[3]{510}\)

a. Find the critical points of the following functions on the domain or on the given interval. b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither. $$f(x)=x /\left(x^{2}+1\right)$$