Question

How do you find all values of c that satisfy the Mean Value Theorem for integrals for `f(x)=x^4` on the interval [-4,4]?

Answer 1

Solve `f'(x) = (f(4)-f(-4))/(4-(-4))`

Explanation:

Discard any solution outside the interval `(-4,4)`.

The conclusion is:

There is a `c` in `(a.b)` such that `f'(c) = (f(b)-f(a))/(b-a)`.

To find the `c`'s, find `f'(x)`,
find `f(b)` and #f(a) then find (f(b)-f(a))/(b-a) solve the equation #f'(x) = (f(b)-f(a))/(b-a)# keeping only the solutions in #(a.b)#.