Chapter 4.4, Problem 14E

a.

To determine

To show: The first and second derivative of the function $f\left(x\right)=\frac{1}{3x^4+5}$ is $f^{\prime }\left(x\right)=-\frac{12x^3}{{\left(3x^4+5\right)}^2}$ and $f^{″}\left(x\right)=\frac{180x^2\left(x^2+1\right)\left(x+1\right)\left(x-1\right)}{{\left(3x^4+5\right)}^3}$ and find the critical points and possible inflection points.

b.

To determine

To find: The intervals such that the function $f\left(x\right)=\frac{1}{3x^4+5}$ is increasing or decreasing.

c.

To determine

To find: The intervals such that the function $f\left(x\right)=\frac{1}{3x^4+5}$ is concave up or concave down.

d.

To determine

To identify: The local extreme values and inflection points of f.

e.

To determine

To state: The x-and y- intercepts of the graph f.

f.

To determine

To state: The asymptotes of the graph f.

g.

To determine

To sketch: The graph of the polynomial $f\left(x\right)=\left(x-3\right){\left(x+3\right)}^2$ and make a complete graph of f.