Let f(x) = 6x^3 + 9. Find the open intervals on which f is...

Let $f(x) = 6x^3 + 9$. Find the open intervals on which $f$ is increasing (decreasing). Then determine the $x$-coordinates of all relative maxima (minima). 1. $f$ is increasing on the intervals 2. $f$ is decreasing on the intervals 3. The relative maxima of $f$ occur at $x = $ 4. The relative minima of $f$ occur at $x = $ Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word \"none\". In the last two, your answer should be a comma separated list of $x$ values or the word \"none\".

Transcript

00:01 The first thing i like to make my students do is understand the behavior of a cubic function.

00:09 So just to give you an idea, we have an odd function with a negative leading coefficient.

00:16 So i already know that this graph needs to end down into the right.

00:20 I'm not sure exactly what happens in between.

00:22 I just kind of lamely put that on.

00:24 Plus, we have to have a y intercept of 9.

00:27 So i don't know exactly what this graph does.

00:31 But what i can do is i can find the derivative of this, which is a derivative of 9 is 0, so don't need to write that down.

00:38 Derivative of 6x is 6.

00:40 And going through the power rule, you bring the 3 in front, subtract 1 from the exponent.

00:46 And if i set that equal to 0, i can find the critical numbers.

00:49 So i'm going to add 3x squared over.

00:51 I'm going to divide the 3 over.

00:54 So x squared equals 1 .5.

00:57 Nope, that equals 2.

00:58 I once did that wrong.

01:01 And then when you square root that, you get positive root 2.

01:05 So because i know the behavior of the graph, i know that we're going to decrease from negative infinity to negative root 2, as well as from positive root 2 to infinity.

01:21 I should have also wrote the increasing first because i think that's the first thing you want.

01:25 It's increasing from negative root 2 to positive root 2...

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