Section 4.5 : The Shape of a Graph, Part I
7. For \(y = 2{x^3} - 10{x^2} + 12x - 12\) answer each of the following questions.
- Identify the critical points of the function.
- Determine the intervals on which the function increases and decreases.
- Classify the critical points as relative maximums, relative minimums or neither.
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a Identify the critical points of the function. Show SolutionWe need the 1st derivative to get the critical points so here it is.
\[\frac{{dy}}{{dx}} = 6{x^2} - 20x + 12\]Now, recall that critical points are where the derivative doesn’t exist or is zero. Clearly this derivative exists everywhere (it’s a polynomial….) and because the derivative can’t be factored in this case we’ll need to do a quick quadratic formula to find where the derivative is zero. The critical points of the function are,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = \frac{{5 \pm \sqrt 7 }}{3} = 0.78475,\,\,2.54858}}\]b Determine the intervals on which the function increases and decreases. Show Solution
To determine the increase/decrease information for the function all we need is a quick number line for the derivative. Here is the number line.
From this we get the following increasing/decreasing information for the function.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{\mbox{Increasing : }}\left( - \infty ,\frac{5 - \sqrt 7}{3} \right)\,\,\,\,\& \,\,\,\left( \frac{5 + \sqrt 7}{3} ,\infty \right)\hspace{0.25in}\hspace{0.25in}{\mbox{Decreasing : }}\,\,\left( \frac{5 - \sqrt 7}{3},\frac{5 + \sqrt 7}{3} \right)}}\]c Classify the critical points as relative maximums, relative minimums or neither. Show Solution
With the increasing/decreasing information from the previous step we can easily classify the critical points using the 1st derivative test. Here is classification of the functions critical points.
\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}x & = \frac{5 - \sqrt 7}{3} = 0.78475\,\,\,\,\,\,:\,{\mbox{ Relative Maximum}}\\ x & = \frac{5 + \sqrt 7}{3} = 2.54858\,\,\,\,\,\,:\,{\mbox{ Relative Minimum}}\end{align*}}\]