#### To determine

**To show:**

f takes on the value 4 at least once on the interval 1,3..

#### Answer

In the following we show that f takes on the value 4 at least once on the interval 1,3.

#### Explanation

**1) Concept:**

Use the mean value theorem for integrals.

**2) Mean Value Theorem for Integrals**

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-a∫abfx dx

that is,

∫abfx dx=fcb-a

**3) Given:**

∫13fx dx=8

**4) Calculation:**

Since f is continuouson 1,3, by using the mean value theorem for integrals, there exists a number c in 1,3 such that

fc=13-1∫13fx dx

Simplifying,

fc=12∫13fx dx

2fc=∫13fx dx

Using the given information,

2fc=8

Divide by 2.

fc=4

**Conclusion:**

So f takes on the value 4 at least once on the interval 1,3.