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

Suppose is a differentiable increasing function for all x. Which number is the larger and why? or ?

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 or ?

Thank you!

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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?