# A Suspected Mean Value Problem

I'm working on the following problem:

Let $h$ be a differentiable function defined on the interval $[0, 3]$, and assume that $h(0) = 1, h(1) = 2$, and $h(3) = 2$. Argue that $h'(x) = 1/4$ at some point in the domain.

I've tried using mean value property and the generalized mean value by guessing some other function. I can obtain that $h'$ will attain the values $0$ and $1/3$. I was going to argue by IVT that $1/4$ will be attained, but I don't think I can assume the derivative is continuous -- so I'm not sure how to proceed.