91Ó°ÊÓ

Identify the open intervals on which the function is increasing or decreasing. $$ y=\frac{x^{3}}{4}-3 x $$

Find two positive numbers that satisfy the given requirements. The second number is the reciprocal of the first and the sum is a minimum.

Numerical and Graphical Analysis In Exercises 3-8, use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{8 x}{\sqrt{x^{2}-3}} $$

Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=x^{4}-2 x^{2} $$

Sse the information to evaluate and compare \(\Delta y\) and \(d y\). $$ y=x^{4}+1 \quad x=-1 \quad \Delta x=d x=0.01 $$

Find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=x(x-3) $$

Determine the open intervals on which the graph is concave upward or concave downward. \(h(x)=x^{5}-5 x+2\)

In Exercises \(7-14,\) find the differential \(d y\) of the given function. $$ y=3 x^{2}-4 $$

Find the two \(x\) -intercepts of the function \(f\) and show that \(f^{\prime}(x)=0\) at some point between the two \(x\) -intercepts. $$ f(x)=x \sqrt{x+4} $$

Determine the open intervals on which the graph is concave upward or concave downward. \(y=2 x-\tan x, \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)