Chapter 4.4, Problem 29E

a.

To determine

To find: To find the sign changes and local extrema.

Answer to Problem 29E

The local extrema are nil.

Explanation of Solution

Given:

The function $f(x)=a{x}^3+b{x}^2+cx+d,a\ne 0$ .

Calculation:

Here,

  $\begin{array}{c}f(x)=a{x}^3+b{x}^2+cx+d,a\ne 0\\ f\text{'}(x)=3a{x}^2+2bx+c\end{array}$

Here $f\text{'}(x)$ is a quadratic polynomial, so there are may be $0,1,2$ Zeros. If it has $0,1$ zeros its sign never changes so $f(x)$ has no local extrema.

If it has $2$ Zeros its sign changes and $f(x)$ has two local extrema.

b.

To determine

To find: To find the sign changes and local extrema.

Answer to Problem 29E

The local extrema are two.

Explanation of Solution

Given:

The function $f(x)=x^3-3x$ .

Calculation:

Here

  $\begin{array}{c}f(x)=x^3-3x\\ f\text{'}(x)=3x^2-3\\ f\text{'}(x)=3(x^2-1)\\ f\text{'}(x)=3(x+1)(x-1)\\ f\text{'}(x)=0\\ x=1\\ x=-1\end{array}$

Here $f\text{'}(x)$ is a quadratic polynomial, so there are may be $0,1,2$ Zeros.

If it has $2$ Zeros its sign changes and $f(x)$ has two local extrema.