Q. 5 - Calculus

Question

Fill in the blanks to complete each of the following theorem statements: 

5. The Mean Value Theorem for Integrals: If $f$is ____ on $[a,b]$, then there is some $c\in \_\_\_$ such that $f\left(c\right)=\_\_\_$

Step-by-Step Solution

Answer

If $f$ is continuous on  $[a,b]$, then there is some $c\in (a,b)$ such that

$f\left(c\right)=\frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx$

Step 1. Given data

We have to fill the blank space,

The Mean Value Theorem for Integrals: If $f$ is ____ on $[a,b]$, then there is some $c\in \_\_\_\_$ such that  $f\left(c\right)=\_\_\_\_$

Step 1. Fill in the blacks

If $f$ is continuous on a $\left[a,b\right]$, then there is some $c\in (a,b)$ such that $f\left(c\right)=\frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx$