Using derivatives to identify extreme values

Suppose that $g$ is a differentiable function and $g'(2) = 0$. In addition, suppose that on $1 \lt x\lt 2$ and $2 \lt x \lt 3$ it is known that $g'(x)$ is positive.

1. Does $g$ have a local maximum, local minimum, or neither at $x = 2$? Why?

2. Suppose that $g''(x)$ exists for every $x$ such that $1 \lt x \lt 3$. Reasoning graphically, describe the behavior of $g''(x)$ for $x$-values near $2$.

3. Besides being a critical number of $g$, what is special about the value $x = 2$ in terms of the behavior of…

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…