022 Exam 1 Sample A, Problem 4

 Problem 4.  Determine the intervals where the function  $h(x)=2x^{4}-x^{2}$ is increasing or decreasing.

Foundations:
When a first derivative is positive, the function is increasing (heading uphill). When the first derivative is negative, it is decreasing (heading downhill). When the first derivative is $0$, it is not quite so clear. If $f'(z)=0$ at a point $z$, and the first derivative splits around it (either $f'(x)<0$  for $x<z$ and $f'(x)>0$  for $x>z$, or $f'(x)>0$  for $x<z$ and $f'(x)<0$  for $x>z$), then the point $(z,f(z))$ is a local maximum or minimum, respectively, and is neither increasing or decreasing at that point.
On the other hand, if the first derivative does not split around $z$, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals. For example, $g(x)=x^{3}$ has the derivative $g'(x)=3x^{2}$. Thus, $g'(0)=0$, but is strictly positive every else. As a result, $g(x)=x^{3}$  is increasing on $(-\infty ,\infty )$.

 Solution:

Find the Roots of the First Derivative:
Note that
$h'(x)\,\,=\,\,8x^{3}-2x\,\,=\,\,2x\left(4x^{2}-1\right)\,\,=\,\,2x(2x+1)(2x-1),$
so the roots of $h'(x)$ are $0$  and $\pm 1/2$.

Make a Sign Chart and Evaluate:
We need to test convenient numbers on the intervals separated by the roots. Using the form
$h'(x)\,\,=\,\,2x(2x+1)(2x-1),$
we can test at convenient points to find
$f'(-10)=(-)(-)(-)=(-),\quad f'(-1/4)=(-)(+)(-)=(+),$
$f'(1/4)\,\,=(+)(+)(-)=(-),\quad f'(10)=(+)(+)(+)=(+).$
From this, we can build a sign chart:
$x:$ $x<-1/2$ $x=-1/2$ $-1/2<x<0$ $x=0$ $0<x<1/2$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=1/2} $x>1/2$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x):} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (+) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (+) }
Notice that at each of our roots, the derivative does split (changes sign as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} passes through each root of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h'(x)} ), so the function is neither increasing or decreasing at each root. Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)} is increasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1/2,0)\cup(1/2,\infty)} , and decreasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty,-1/2)\cup(0,1/2)} .

Final Answer:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)} is increasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1/2,0)\cup(1/2,\infty)} , and decreasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty,-1/2)\cup(0,1/2)} .

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