Mean Value Theorem

Let $f$ be a function that satisfies the hypothesis:

1. $f$ is continuous on the closed interval $[a,b]$
2. $f$ is differentiable on open interval $(a,b)$ Then there is a number $c$ in $(a,b)$ such that: $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$ This can be used to find the mean value of the function

Proof

• Mean Value Theorem Proof

Problems

• Find smallest possible value for f(4) given f(1) and f’(x) > 2

Corollary

• Constant Function Corollary
• Antiderivative Corollary