Question

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]?

Answer 1

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}`.