How would I go about showing this question:
Suppose that the function $f:[0,1] \rightarrow [0,1]$ is continuous. Use the Intermediate Value Theorem to prove that there exists $c \in [0,1]$ such that $$f(c) = c(2-c^2)$$
My thoughts are we can define a function say $g(x) := f(x) - x(2-x^2)$ and then $g(0) = f(0) $and then $g(1) = f(1) - 1 $ and then I'm stuck on how to complete it to get to the point where
$$g(0) < 0 < g(1)$$ so there would exist a c s.t. $g(c) = 0$ [by IVT] and thus $f(c) = c(2-c^2)$
If anyone could help, would be much appreciated thank you!