Exercise 13

Let $p$ be a function whose second derivative is $p''(x) = (x+1)(x-2)e^{-x}$.

1. Construct a second derivative sign chart for $p$ and determine all inflection points of $p$.

2. Suppose you also know that $x = \frac{\sqrt{5}-1}{2}$ is a critical number of $p$. Does $p$ have a local minimum, local maximum, or neither at $x = \frac{\sqrt{5}-1}{2}$? Why?

3. If the point $(2, \frac{12}{e^2})$ lies on the graph of $y = p(x)$ and $p'(2) = -\frac{5}{e^2}$, find the equation of the tangent line to $y = p(x)$ at the point where $x = 2$. Does the tangent line lie above the curve, below the curve, or neither at this value? Why?