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 $f(4)+f'(4)\Delta x$?