I came across the following exercise in Spivak:
Suppose that $f(x)$ is a differentiable function and suppose that $f'(x)$ is increasing. Show that every tangent line of $f(x)$ intersects the graph only once.
My intuition is as follows:
Consider the tangent line $t(x)$ for any point $a$. Clearly, we have $t'(a) = f'(a)$. Since $f'(x)$ is an increasing function whereas $t'(x)$ is constant, the two graphs can never intersect again for $x\geq a$ because $f(x)$ "grows faster" than $t(x)$.
How do make this argument precise?
Best Answer
Consider the tangent $l$ to $f(x)$ at $x=a$. This has gradient $f'(a)$. Let $c>a$. Then by the mean value theorem, there is some $a<b<c$ such that $f'(b)(c-a)=f(c)-f(a)$. This means $f(c)=f(a)+f'(b)(c-a)>f(a)+f'(a)(c-a)$ which is the value of $l$ at $c$. Do the same for the case $c<a$.