Which number is larger if $f'(x)$ is a differentiable increasing function for all $x$?

Question

Suppose $f'(x)$ is a differentiable and increasing function for all $x$. Which number is the largest and why? $f(4+\Delta x)$ or $f(4)+f'(4)\Delta x$?

I believe $f(x)$ must be concave up everywhere since the derivative is increasing, but I am not sure how to figure out which of those is larger based on that. Would the tangent like approximation be $f(4+\Delta x)$ or $f(4)+f'(4)\Delta x$?

Thank you!

Answer 1

The tangent line approximation at $x=4$ is $f(4)+f'(4)(x-4)$. You probably mean $x-4$ by $\Delta x$. It cannot be $f(4+\Delta x)$ as this can not be linear-a straight line does not have increasing derivative.

You are right that $f(x)$ is concave up everywhere. Does that mean it is above or below the tangent line?