91Ó°ÊÓ

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\).$$ f(x)=x^{2 / 3}-1,[-8,8] $$

Identify the open intervals on which the function is increasing or decreasing. $$ y=x \sqrt{16-x^{2}} $$

Find the point on the graph of the function that is closest to the given point. $$ \frac{\text { Function }}{f(x)=\sqrt{x}} \quad \frac{\text { Point }}{(4,0)} $$

Find any critical numbers of the function. $$ \begin{array}{l} h(x)=\sin ^{2} x+\cos x \\ 0

In Exercises \(11-14,\) find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{2}-4}\) (b) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{3 / 2}-4}\) (c) \(\lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x-4}\)

Find the differential \(d y\) of the given function. $$ y=\frac{1}{3} \cos \left(\frac{6 \pi x-1}{2}\right) $$

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=\frac{x}{x^{2}+1}\)

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=3-|x-3|,[0,6] $$

Find the differential \(d y\) of the given function. $$ y=\arctan (x-2) $$

Find the point on the graph of the function that is closest to the given point. $$ \frac{\text { Function }}{f(x)=\sqrt{x}-8} \quad \frac{\text { Point }}{(2,0)} $$