022 Exam 1 Sample A, Problem 4

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Determine the intervals where the function  is increasing or decreasing.

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 it is not quite so clear. If   at a point , and the first derivative splits around it (either   for and   for , or   for and   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} ), then the point 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 , then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals. For example, has the derivative . Thus, , but is strictly positive everywhere else. As a result,   is increasing on .


Find the Roots of the First Derivative:  
Note that
so the roots of are   and .
Make a Sign Chart and Evaluate:  
We need to test convenient numbers on the intervals separated by the roots. Using the form
we can test at convenient points to find
From this, we can build a sign chart:

Notice that at each of our roots, the derivative does split (changes sign as passes through each root of ), so the function is neither increasing or decreasing at each root. Thus, is increasing on , and decreasing on .
Final Answer:  
is increasing on , and decreasing on .

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