Midpoint Versus Secant

Let $f(x) = \mathrm{e}^x$ for $x \in \mathbb{R}$.

1. Find the tangent line to the graph of the inverse function of $f(x)$ at the point $(1, 0)$. 2. Prove that the curves $y = f(x)$ and $y = \dfrac{1}{2}x^2 + x + 1$ have exactly one common point. 3. For $a < b$, compare

and justify your answer.