91Ó°ÊÓ

Find the two numbers whose sum is 50 and whose product is a maximum.

Use a graphing calculator to estimate the \(x\) -coordinates of the inflection points of each function, rounding your answers to two decimal places. \(f(x)=x^{5}-3 x^{3}+6 x+2\)

Sketch the graph of each rational function after making a sign diagram for the derivative and finding all relative extreme points and asymptotes. \(f(x)=\frac{6 x}{x^{2}+9}\)

Show that the largest rectangle with a given perimeter is a square.

\(X\) and \(y\) are functions of \(t\) Differentiate with respect to \(t\) to find a relation between \(d x / d t\) and \(d y / d t\). \(x y^{2}=96\)

Sketch the graph of each rational function after making a sign diagram for the derivative and finding all relative extreme points and asymptotes. \(f(x)=\frac{3}{x^{2}-1}\)

Sketch the graph of a function \(f(x)\) that satisfies the stated conditions. Mark any inflection points by writing IP on your graph. a. \(f\) is continuous and differentiable everywhere. b. \(f(0)=3\) c. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) and \((0, \infty)\) d. \(f^{\prime}(x)<0 \quad\) on \((-4,0)\) e. \(f^{\prime \prime}(x)<0 \quad\) on \((-\infty,-2)\) f. \(f^{\prime \prime}(x)>0\) on \((-2, \infty)\)

\(X\) and \(y\) are functions of \(t\) Differentiate with respect to \(t\) to find a relation between \(d x / d t\) and \(d y / d t\). \(3 x^{2}-7 x y=12\)

\(X\) and \(y\) are functions of \(t\) Differentiate with respect to \(t\) to find a relation between \(d x / d t\) and \(d y / d t\). \(2 x^{3}-5 x y=14\)

Sketch the graph of a function \(f(x)\) that satisfies the stated conditions. Mark any inflection points by writing IP on your graph. a. \(f\) is continuous and differentiable everywhere. b. \(f(0)=4\) c. \(f^{\prime}(x)<0\) on \((-\infty,-1)\) and \((3, \infty)\) d. \(f^{\prime}(x)>0\) on \((-1,3)\) e. \(f^{\prime \prime}(x)>0\) on \((-\infty, 1)\) f. \(f^{\prime \prime}(x)<0 \quad\) on \((1, \infty)\)