Let f(x) = 2x^3 + 9. Find the open intervals on which f is increasing (decreasing). Then...

Question:

Let {eq}f(x) = 2x^3 + 9 {/eq}. Find the open intervals on which {eq}f {/eq} is increasing (decreasing). Then determine the {eq}x {/eq}-coordinates of all relative maxima (minima).

1) {eq}f {/eq} is increasing on the intervals: _____

2) {eq}f {/eq} is decreasing on the intervals: _____

3) The relative maxima of {eq}f {/eq} occur at {eq}x = {/eq} _____. The relative minima of {eq}f {/eq} occur at {eq}x = {/eq} _____.

Increasing/Decreasing/Maxima/Minima:

A function is said to be increasing on an interval if the derivative is positive in that interval.

A function is said to be decreasing on an interval if the derivative is negative in that interval.

If the derivative of the function changes signs through x=a and it is continuous there, then it has a relative minimum or a maximum there.

Answer and Explanation:

Note that {eq}f'(x) = 6x^2 {/eq}

Now, {eq}f'(x) > 0 \Rightarrow x \in \mathbb R. {/eq}

That is, the graph is always increasing. That is, it has no relative maxima or minima.

1) {eq}f {/eq} is increasing on the intervals: {eq}(-\infty, +\infty). {/eq}

2) {eq}f {/eq} is decreasing on the intervals: none

3) The relative maxima of {eq}f {/eq} occur at {eq}x = {/eq} no solution.

4) The relative minima of {eq}f {/eq} occur at {eq}x = {/eq} no solution.


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Finding Minima & Maxima: Problems & Explanation

from General Studies Math: Help & Review

Chapter 5 / Lesson 2
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