First Derivative Test Exercises

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Here we’ll practice using the first derivative test.

The function $f(x) = x^3-6x+1$ has two critical points. If we call these critical point $a$ and $b$ , and order them such that $a < b$ , then

$a = \answer {-\sqrt {2}}$

$b=\answer {\sqrt {2}}$

On $(\infty ,a)$ , $f$ is

increasing decreasing

On $(a,b)$ , $f$ is

increasing decreasing

On $(b,\infty )$ , $f$ is

increasing decreasing

Consider $f$ restricted to the domain $[-3,2.5]$ .

On this interval, the absolute maximum value of $f$ is $\answer {4\sqrt {2}+1}$ .

On this interval, the absolute minimum value of $f$ is $\answer {-8}$ .

Consider the function $f(x) =\displaystyle \frac {1}{4\sqrt {x}}+x$ .

What is the domain of this function?

Domain: $(\answer {0}, \infty )$

$f(x)$ has only one critical point. Call this critical point $a$ .

$a = \answer {0.25}$

On $(-\infty ,0)$ , $f$ is

increasing decreasing not defined

On $(0,a)$ , $f$ is

increasing decreasing

On $(a,\infty )$ , $f$ is

increasing decreasing

$f(a)$ is an absolute

maximum minimum

The function $f(x) =\frac {x^2-2x+1}{x(1+x^2)}$ has two critical points and one point where $f(x)$ is undefined. If we call these points $a$ , $b$ , and $c$ and order them such that $a < b$ , then

$a = \answer {0}$

$b=\answer {1}$

$c=3.383$

On $(-\infty ,a)$ , $f$ is

increasing decreasing

On $(a,b)$ , $f$ is

increasing decreasing

On $(b,c)$ , $f$ is

increasing decreasing

On $(c,\infty )$ , $f$ is

increasing decreasing

$x=a$ is the location of a

Local max Local Min Neither

$x=b$ is the location of a

Local max Local Min Neither

$x=c$ is the location of a

Local max Local Min Neither

If $f'(x) =\frac {x^2+2x-3}{x+1}$ then the function $f(x)$ has two critical points and one point where the function is undefined. If we call these points $a$ , $b$ , and $c$ and order them such that $a < b < c$ , then

$a = \answer {-3}$

$b=\answer {-1}$

$c=\answer {1}$

On $(-\infty ,a)$ , $f$ is

increasing decreasing

On $(a,b)$ , $f$ is

increasing decreasing

On $(b,c)$ , $f$ is

increasing decreasing

On $(c,\infty )$ , $f$ is

increasing decreasing

$x=a$ is the location of a

Local max Local Min Neither

$x=b$ is the location of a

Local max Local Min Neither

$x=c$ is the location of a

Local max Local Min Neither

The function $f(x) = 3x^4-8x^3+6x^2+7$ has two critical points. If we call these critical points $a$ and $b$ , and order them such that $a < b$ , then

$a = \answer {0}$

$b=\answer {1}$

On $(-\infty ,a)$ , $f$ is

increasing decreasing

On $(a,b)$ , $f$ is

increasing decreasing

On $(b,\infty )$ , $f$ is

increasing decreasing

$x=a$ is the location of a

Local max Local Min Neither

$x=b$ is the location of a

Local max Local Min Neither