I've tried several ways of answering this type of question but I can't seem to figure it out. This is one variation of it:
Suppose that $f(x)$ is differentiable on [38,50]. Also suppose that $f(50)=21$ and $f'(x)<=10$ for all $x\in[38,50]$. What is the least possible value that $f(38)$ can take?
I have concluded that this is rooted in the Extreme-Value theorem but I'm not totally sure how to use that here.
My attempt to answer this: $f'(x)<=10$ -- integrates to $f(x)<=10x$ this leads me to conclude that the highest value that f(38) can take is 380.
Which is not what the question is asking me.
If you can help that would be splendid.