Chapter 2.2, Problem 7E

To determine

The condition on the polynomial $f\left(x\right)$ whose graph is given here so that according to Intermediate Value Theorem the polynomial has a zero in interval $\left[x_1,x_2\right]$ .

Answer to Problem 7E

The required condition is that $f\left(x_2\right)>0$ .

Explanation of Solution

Given data:

The graph of polynomial function $f\left(x\right)$ is shown here.

  

Concept used:

According to Intermediate Value Theorem the polynomial $f\left(x\right)$ has a zero in interval $\left[x_1,x_2\right]$ , iff $f\left(x_1\right)$ and $f\left(x_2\right)$ are of opposite signs.

  

From the graph $f\left(x_1\right)<0$ therefore according to Intermediate Value Theorem the polynomial $f\left(x\right)$ has a zero in interval $\left[x_1,x_2\right]$ if $f\left(x_2\right)>0$ .

Thus, for a zero of polynomial function $f\left(x\right)$ in the interval $\left[x_1,x_2\right]$ other required condition is $f\left(x_2\right)>0$ .

Conclusion:

The required condition is that $f\left(x_2\right)>0$ .