Calculus (Derivatives) prove the given question Let $f(x)$ and $g(x)$ be two continuous and differentiable functions defined on the interval $[0,2]$. Suppose $f(0)&gt;g(0)$ and $f(1)&lt;g(1)$, show that there exists a number $u$ in the interval $(0,1)$ such that $f(u)=g(u)$. Suppose further that $f(2)&gt;g(2)$. Show that there is a number $w$ in the interval $(0,2)$ such that $f'(w)=g'(w)$. (Hint: consider $h(x)=f(x)-g(x)$.)


I haven't really made any progress with this question. I did try applying the Intermediate value theorem, Rolle's theorem and the Mean value theorem but I'm not very sure how to go about it also I don't get why $h(x)$ would be $f(x) - g(x)$. Any hints would be appreciated!
Since the function f(x) is first increasing and then decreasing we get that it must intersect g(x) somewhere between 0 and 1. Also it becomes a monotonous function in between 0 and 2 hence there must be at least one point where the derivates(slopes) are equal hence the result
Since the function f(x) is first increasing and then decreasing we get that it must intersect g(x) somewhere between 0 and 1. Also it becomes a monotonous function in between 0 and 2 hence there must be at least one point where the derivates(slopes) are equal hence the result