91Ó°ÊÓ

Determine the function \(\mathrm{f}\) satisfying the given conditions. $$ f^{\prime}(x)=\frac{1}{x}, f(e)=-3 $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ k(x)=\sin x-\frac{\sqrt{3}}{2} x $$

Find all extreme values (if any) of the given function on the given interval. Determine at which numbers in the interval these values occur. $$ g(x)=1 / x ;(0,3] $$

Use Rolle's Theorem to show that there is a solution of the equation \(\tan x=1-x\) in \((0,1)\). (Hint: Let \(f(x)\) \(=(x-1) \sin x\), and find \(f(0), f(1)\), and \(\left.f^{\prime}(x) .\right)\)

Find all inflection points (if any) of the graph of the function. Then sketch the graph of the function. $$ f(x)=(x+2)^{3} $$

Determine all functions \(f\) satisfying the given conditions. $$ f^{\prime \prime}(x)=0 \text { (Hint: Use Theorem 4.6 twice.) } $$

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ k(x)=\sin \left(\frac{x^{2}}{1+x^{2}}\right) $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{1}{\ln x} $$

Find all extreme values (if any) of the given function on the given interval. Determine at which numbers in the interval these values occur. $$ f(t)=-1 /(2 t) ;(0, \infty) $$

Use Rolle's Theorem to show that there is a solution of the equation \(\cot x=x\) in \((0, \pi / 2) .\) (Hint: Let \(f(x)=x \cos x\) for in \([0, \pi / 2] .)\)