022 Exam 1 Sample A, Problem 4

 Problem 4.  Determine the intervals where the function  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)=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 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,} it is not quite so clear. If 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 f'(z)=0}   at a point 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 z} , and the first derivative splits around it (either 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 f'(x)<0}   for 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<z} and 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 f'(x)>0}   for $x>z$, or $f'(x)>0$  for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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:
$x:$ $x<-1/2$ $x=-1/2$ $-1/2<x<0$ $x=0$ $0<x<1/2$ $x=1/2$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x>1/2} $f'(x):$ $(-)$ $0$ $(+)$ $0$ $(-)$ $0$ $(+)$
Notice that at each of our roots, the derivative does split (changes sign as $x$ passes through each root of $h'(x)$), so the function is neither increasing or decreasing at each root. Thus, $h(x)$ is increasing on $(-1/2,0)\cup (1/2,\infty )$, and decreasing on $(-\infty ,-1/2)\cup (0,1/2)$.

Final Answer:
$h(x)$ is increasing on $(-1/2,0)\cup (1/2,\infty )$, 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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