Chapter 2.3, Problem 49E

Solution

To find the zeroes of the function $f\left(x\right)=x^3-36x$ algebraically or graphically.

The zeroes of the function $f\left(x\right)=x^3-36x$ are:

  $x=-6$ , $x=0$ and $x=6$ .

Given:

  $f\left(x\right)=x^3-36x$ .

Concept Used:

Zeroes of a function:

Given a function $f\left(x\right)$ . A point $x$ is said to be a zero of $f$ if and only if $f\left(x\right)=0$ .

Now, $f\left(x\right)=0$ if and only if $\left(x,0\right)$ is an $x$ -intercept of $f$ .

Thus, $x$ is a zero of $f$ if and only if $\left(x,0\right)$ is an $x$ -intercept of $f$ .

The function $f\left(x\right)=x^3-36x$is graphed along with the $x$ intercepts:

  

The function $f\left(x\right)=x^3-36x$ is coloured in red and the intercepts are coloured in grey.

Find the zeroes of $f\left(x\right)=x^3-36x$ from the graph:

It is clear from the graph that the function has only three $x$ -intercepts:

  $(-6,0)$ , $(0,0)$ and $(6,0)$ .

Thus, the zeroes of the function $f\left(x\right)=x^3-36x$ are:

  $x=-6$ , $x=0$ and $x=6$ .

Conclusion:

The zeroes of the function $f\left(x\right)=x^3-36x$ are:

  $x=-6$ , $x=0$ and $x=6$ .