Worksheet for Section 4.1

Secton 4.1 is about using the first and second derivatives of a function to determine characteristics of the function's graph, in particular where a function takes maximum or minimum values. Start by reviewing what you have already seen about the information that first and second derivatives of a function tell you about the graph of the function:

• if $f'(x)>0$ on an interval $I$, then $f(x)$ is increasing on $I$.
• if $f'(x)<0$ on an interval $I$, then $f(x)$ is decreasing on $I$.
• if $f''(x)>0$ on an interval $I$, then the graph of $f(x)$ is concave up on $I$.
• if $f''(x)<0$ on an interval $I$, then the graph of $f(x)$ is concave down on $I$.

For example, look at the function $f(x)=x^3-3x^2$. First, use the derivatives to determine where $f(x)$ is increasing or decreasing, and where it is concave up or concave down. Use this information (possibly along with values of $f(x)$ at a few interesting places) to sketch the graph of $f(x)$.

The strategy for finding maximum and minimum values will involve finding good candidate points, then checking them to see which value is largest and which is smallest. To characterize the best candidate points, consider the ``hills'' and ``valleys'' in the graph of a function -- defined more formally:

• If there is an open interval $I$ containing $c$ so that $f(c)$ is the maximum value of $f(x)$ on $I$, then $f(c)$ is a local maximum for $f(x)$.
• If there is an open interval $I$ containing $c$ so that $f(c)$ is the minimum value of $f(x)$ on $I$, then $f(c)$ is a local minimum for $f(x)$.

For example, look again at the function $f(x)=x^3-3x^2$ above, and determine whether it has a local maximum or a local minimum. Does $g(x)=\vert x\vert$ have a local maximum or a local minimum? The behavior of these two functions at their local extreme values is important, and provides a good way of finding candidate points for extreme values:

• Suppose $f(x)$ is defined at $x=c$. If $f'(c)=0$ or $f'(x)$ is undefined at $x=c$, then $c$ is a critical number for $f(x)$. The point $(c,f(c))$ is a critical point; the value $f(c)$ is a cricical value.

Critical numbers can be found from the derivative of the function, then -- for example, find the critical numbers of $f(x)=x^3-3x^2$, by solving the equation $f'(x)=3x^2-3=0$ for $x$. How can you find the critical number(s) for $g(x)=\vert x\vert$?

This behavior of functions at critical numbers provides the last part of the technique for finding extreme values. The important link is between local extreme values and critical numbers:

• Fact: If $f(x)$ takes a local maximum value or a local minimum value at $x=c$, then $c$ is a critical number for $f(x)$

Some functions have critical numbers in places where they don't have local extreme values, though -- for example, $h(x)=x^3+1$ has a critical number at $x=0$, but this is not a relative extreme value. Here are some examples -- in each case, find the critical values and determine from the graphs of the functions for each critical value whether it is a local maximum, a local minimum, or neither:

$$f(x)=\frac{x^2-1}{x^3}\qquad\qquad g(x)=x^4-2x^2\qquad\qquad h(x)=2x-3x^{2/3}$$

Determine the intervals where each of the following functions are increasing or decreasing:

$$f(x)=x^3-3x\qquad\qquad g(x)=\frac1{x^2+1}$$

This information about where $f(x)$ is increasing or decreasing has another nice use, for finding local maximum and minimum points -- this is the First Derivative Test. Suppose $c$ is a critical number for $f(x)$, where $f(x)$ is continuous on some open interval $I$ containing $c$, and differentiable on $I$, except possibly at $c$; then the value of $f(c)$ can be described using the first derivative $f'(x)$ as follows:

• If $f'(x)$ changes sign from negative to positive (decreasing to increasing) at $c$, then $f(c)$ is a local minimum for $f(x)$.
• If $f'(x)$ changes sign from positive to negative (increasing to decreasing) at $c$, then $f(c)$ is a local maximum for $f(x)$.
• If $f'(x)$ does not change sign at $c$, then $f(c)$ is neither a local maximum nor a local minimum.

Since you already have this sort of information for the two functions above, determine where each of the functions take relative maximum and relative minimum values using this First Derivative Test. Now repeat the process for the following two funtions:

$$f(x)=(x^2-4)^{2/3}\qquad\qquad g(x)=\frac{x}2-\sin x$$

The Second Derivative Test uses information about concavity from $f''(x)$ to classify critical numbers -- if $c$ is a critical number for $f(x)$ and $f''(c)$ is defined (so that $f(x)$ must be continuous and twice differentiable on some open interval containing $c$), then:

• If $f''(c)>0$, then $f(c)$ is a local minimum for $f(x)$.
• If $f''(c)<0$, then $f(c)$ is a local maximum for $f(x)$.

(If $f''(x)=0$, then the Test fails.) Apply this test wherever you can to the functions below -- first, you will have to find their critical numbers. Use this information to help you sketch graphs of the functions. (Several of these functions are the same as other functions above.)

$$f_1(x) =x^2 g_1(x) =x^3 h_1(x) =x^3-3x$$

$$f_2(x) =\frac1{x^2+1} g_2(x) =\frac{2}{x^2-4} h_2(x) =\frac{x}{x^2+1}$$

$$f_3(x) =e^{-x^2} g_3(x) =\sin x h_3(x) =\tan^{-1} x$$