Chapter 2.3, Problem 44E

Solution

To graph the function $f\left(x\right)=-x^3+3x^2+7x-2$ and to approximate all of its zeroes.

The zeroes of the function $f\left(x\right)=-x^3+3x^2+7x-2$ are approximately:

  $x=-1.73$ , $x=0.26$ and $x=4.47$ .

Given:

  $f\left(x\right)=-x^3+3x^2+7x-2$ .

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 $f\left(x\right)=-x^3+3x^2+7x-2$is graphed along with the $x$

intercepts:

  

The function $f\left(x\right)=-x^3+3x^2+7x-2$ is coloured in red and the intercepts are coloured in grey.

Estimate the zeroes of $f\left(x\right)=-x^3+3x^2+7x-2$from the graph:

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

  $(-1.73,0)$ , $(0.26,0)$ and $(4.47,0)$ .

Thus, the zeroes of the function $f\left(x\right)=-x^3+3x^2+7x-2$ are approximately:

  $x=-1.73$ , $x=0.26$ and $x=4.47$ .

Conclusion:

The zeroes of the function $f\left(x\right)=-x^3+3x^2+7x-2$ are approximately:

  $x=-1.73$ , $x=0.26$ and $x=4.47$ .