§3.3 Using derivatives to identify extreme values (continued)

Summary

This problem concerns a function about which the following information is known: -$f$ is a differentiable function defined at every real number $x$-$f(0) = -1/2$-$y = f'(x)$ has its graph given at center in Figure 3.1

• Construct a first derivative sign chart for $f$. Clearly identify all critical numbers of $f$, where $f$ is increasing and decreasing, and where $f$ has local extrema. - On the right-hand axes, sketch an approximate graph of $y = f''(x)$. - Construct a second derivative sign chart for $f$. Clearly identify where $f$ is concave up and concave down, as well as all inflection points. - On the left-hand axes, sketch a possible graph of $y = f(x)$.

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. - Does $g$ have a local maximum, local minimum, or neither at $x = 2$? Why? - 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 $2$. - Besides being a critical number of $g$, what is special about the value $x = 2$ in terms of the behavior of the graph of $g$?

Suppose that $h$ is a differentiable function whose first derivative is given by the graph in Figure 3.1.

• How many real number solutions can the equation $h(x) = 0$ have? Why? - If $h(x) = 0$ has two distinct real solutions, what can you say about the signs of the two solutions? Why? - Assume that $\lim_{x \to \infty} h'(x) = 3$, as appears to be indicated in Figure 3.1. How will the graph of $y = h(x)$ appear as $x \to \infty$? Why? - Describe the concavity of $y = h(x)$ as fully as you can from the provided information.

Suppose that $g(x)$ is a function continuous for every value of $x \ne 2$ whose first derivative is $g'(x) = \frac{(x+4)(x-1)^2}{x-2}$. Further, assume that it is known that $g$ has a vertical asymptote at $x = 2$. - Determine all critical numbers of $g$. - By developing a carefully labeled first derivative sign chart, decide whether $g$ has as a local maximum, local minimum, or neither at each critical number. Note: observe that $g'(x)$ can change sign at the vertical asymptote of $g(x)$. - Does $g$ have a global maximum? global minimum? Justify your claims. - What is the value of $\lim_{x \to \infty} g'(x)$? What does the value of this limit tell you about the long-term behavior of $g$? - Sketch a possible graph of $y = g(x)$.

Let $p$ be a function whose second derivative is $p''(x) = (x+1)(x-2)e^{-x}$. - Construct a second derivative sign chart for $p$ and determine all inflection points of $p$. - 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? - 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?