91Ó°ÊÓ

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ g(x)=x^{3}-6 x^{2}+2 x+3 $$

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{\theta \rightarrow 0} \frac{\theta+\sin \theta}{\tan \theta} $$

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ h(x)=x^{2}+1 \text { on }(-2,1] $$

Satisfying Postal Requlations Postal regulations specify that a package sent by priority mail may have a combined length and girth of no more than 108 in. Find the dimensions of a cylindrical package with the greatest volume that may be sent by priority mail. What is the volume of such a package?

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ f(t)=t^{4}-2 t^{3} $$

In Exercises 9-16, verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval, and find all values of \(c\) that satisfy the conclusion of the theorem. $$ h(x)=x \sqrt{2 x+1} ; \quad[0,4] $$

Use Newton's method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than \(0.0001\). The zero of \(f(x)=5 x+\cos x-5\) between \(x=0\) and \(x=1\). Take \(x_{0}=0.5\).

Find the limit. $$ \lim _{x \rightarrow 0^{-}} \frac{x+1}{\sqrt{x}(x-1)^{2}} $$

(a) find the intervals on which \(f\) is increasing or decreasing, and (b) find the relative maxima and relative minima of \(\vec{f}\). $$ f(x)=2 x^{3}+3 x^{2}-12 x+5 $$

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 0} \frac{\sin x-x \cos x}{\tan ^{3} x} $$