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 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 $f'(z)=0$ at a point $z$, and the first derivative splits around it (either $f'(x)<0$  for $x<z$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)>0}   for $x>z$, or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=x^{3}} has the derivative $g'(x)=3x^{2}$. Thus, $g'(0)=0$, but is strictly positive every else. As a result, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=x^{3}} is increasing on $(-\infty ,\infty )$.

 Solution:

Find the Derivatives and Their Roots:
Note that

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