If f(x) = |(x^2 − 4)(x^2 + 2)|, how many numbers in the interval [0, 1] satisfy the conclusion of the Mean Value Theorem?

3 answers

since f(x) < 0 in the interval, we can use
f(x) = -(x^2-4)(x^2+2)
f(1) = 9
f(0) = 8
the slope of the secant is this (9-8)/(1-0) = 1
By inspection of the graph, there appear to be two values that work. But to be sure,
f'(x) = -4x(x^2-1)
So, we want to find c such that
-4c(c^2-1) = 1
solve that using your favorite tools and there are indeed two values for c.
Thank you! However, wouldn't there be three values for c?
yes, but one is not included in the interval [0,1]
Take a look at the graph here:

https://www.wolframalpha.com/input/?i=-(x%5E2+%E2%88%92+4)(x%5E2+%2B+2)