Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 199: 56

See explanations.

Step 1. Given a function $f(x)=x^2$, we can find its derivative as $f′(x)=2x$ Step 2. In the interval of $[a,b]$, the Mean Value Theorem gives $f′(c)=\frac{f(b)-f(a)}{b-a}=\frac{b^2-a^2}{b-a}=a+b$ Step 3. Combining the above results, we get $2c=a+b$ which gives $c=\frac{a+b}{2}$