Second Derivative Test Exercises

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Here we’ll practice using the second derivative test.

The function $f(x) = x^3-27x+6$ has two critical points. If we call these critical points $a$ and $b$ , and order them such that $a < b$ , then

$a = \answer {-3}$

$b=\answer {3}$

$f''(a)$ is

positive negative , so $f(a)$ is a local max min .

$f''(b)$ is

positive negative , so $f(b)$ is a local max min .

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

$a = \answer {-1}$

$b=\answer {1}$

$f''(a)$ is

positive negative , so $f(a)$ is a local max min .

$f''(b)$ is

positive negative , so $f(b)$ is a local max min .

Consider the function $f(x) = x\sqrt {4-x}$ .

What is the domain of $f(x)$ ? Domain: $(-\infty , \answer {4}]$

$f(x)$ has only one critical point, $x=a$ .

$a = \answer {8/3}$

$f''(a)$ is

positive negative , so $f(a)$ is a local max min .

The function $f(x) = x^4-4x^3+16x-3$ has two critical points. If we call these critical points $a$ and $b$ , and order them such that $a < b$ , then

$a = \answer {-1}$

$b=\answer {2}$

At $x=a$ , the second derivative test

Indicates a local maximum occurs at $x=a$ Indicates a local minimum occurs at $x=a$ Fails, but the First derivative test indicates $x=a$ is the location of a local max Fails, but the First derivative test indicates $x=a$ is the location of a local min Fails, and the First derivative test indicates that $x=a$ is not the location of a local extrema

At $x=b$ , the second derivative test

Indicates a local maximum occurs at $x=b$ Indicates a local minimum occurs at $x=b$ Fails, but the First derivative test indicates $x=b$ is the location of a local max Fails, but the First derivative test indicates $x=b$ is the location of a local min Fails, and the First derivative test indicates that $x=b$ is not the location of a local extrema

The function $f(x) = x^4-4 x^3+6 x^2-4 x+3$ has only one critical point, $x=a$ .

$a = \answer {1}$

At $x=a$ , the second derivative test