How do you find all numbers c that satisfy the conclusion of the Mean Value Theorem for f(x)= x^3 + x - 1 over [0,2]?

1 Answer
Apr 16, 2015

First, find the derivative: f'(x)=3x^2+1. Next, find the average rate of change of f over the interval [0,2]: \frac{f(2)-f(0)}{2-0}=\frac{10}{2}=5. At this point, set f'(c)=5 and solve for c as follows: 3c^{2}+1=5 so 3c^{2}=4 and c^{2}=\frac{4}{3}. There's one value of c between 0 and 2 that satisfies the conclusion of the Mean Value Theorem: c=\sqrt{4/3}=\sqrt{4}/\sqrt{3}=2/\sqrt{3}=\frac{2\sqrt{3}}{3}.