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Find the local maximum and minimum values of $ f $ using both the First and Second Derivative Tests. Which method do you prefer?

$ f(x) = 1 + 3x^2 - 2x^3 $

\[

f(x)=1+3 x^{2}-2 x^{3} \Rightarrow f^{\prime}(x)=6 x-6 x^{2}=6 x(1-x)

\]

First Derivative Test: $f^{\prime}(x)>0 \Rightarrow 0<x<1$ and $f^{\prime}(x)<0 \Rightarrow x<0$ or $x>1 .$ since $f^{\prime}$ changes from negative

to positive at $x=0, f(0)=1$ is a local minimum value; and since $f^{\prime}$ changes from positive to negative at $x=1, f(1)=2$ is

a local maximum value.

Second Derivative Test: $f^{\prime \prime}(x)=6-12 x, f^{\prime}(x)=0 \Leftrightarrow x=0,1, f^{\prime \prime}(0)=6>0 \Rightarrow f(0)=1$ is a local

minimum value. $f^{\prime \prime}(1)=-6<0 \Rightarrow f(1)=2$ is a local maximum value.

Preference: For this function, the two tests are equally easy.

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Okay, so we're being asked to find the local Macklin men is in the first and second derivative test. And then, ah, assassin, which meant that do you prefer? So I'm just going to tell you both methods, and that's gonna be kind of for you to decide. And I kind of talked a little bit about the advantage of disagreements for both of them. All right, so, using the virtue of this task me at the first take, the first derivative that would take a prime rex. And then this gives us six six my neck. Okay, six X minus six x squared. There's a common factor of sex sex, so you can pull that out. That leaves just one minus X. We said this whole thing is here. So we get X people two zero and one. Then we create a sign chart to figure out how the function is behaving so that we look or have been evaluated at the two at zero and one enjoy, like Freetown. Well, look at the parts of the function six x in one minute thick, and when you multiply together, you get the sign of a prime. So six x will be negative. Positive? Positive. And then this will be positive. Positive? Negative. Now you multiply this through, and then you got positive. Negative, Negative, Positive, positive, positive, positive time. Negative. Negative. So that means so we can look at how the function is acting at this point. So we can say it is decreasing. And then it is increasing. So then, once increasing, decreasing, we have a local men. So we have a local men at tactical zero and then received at the function is increasing, increasing and decreasing. We have a local max at one. Yeah, so that is how we find the local men in math using the first, the first river. Caf. So then, for the second derivative, Kath, what we're going to do is we first take the second derivative and then what we're going to do, it gives us six minus twelve eggs. And I think the best way to do this is Ah. First look at the point in which the where f prime with zeros after I'm a zero X equals zero and X equals one. And, um, this is relevant because ah, this these the reason why they're zero. Because what the sign changes. So that means that in order for this kind of change that to be a point where the slope is equal to zero and so in others for pickles, you're zero on one. So then we take those two point and plug it into our values, our second derivative. So we evaluate at times zero in a crime of one and six months zero six and then this will be six. Minus twelve will be negative. Six. Since of prima's positive, we know that it is Khan Cave up con care, which means look, something like this should mean there's a local men so local then and then if it's this's con cave down because it is negative, complicated down. It looks like this that there's a local Max and that is how you determine the local men's and Mac. So for me, perfect. Personally, I prefer dis method, the first communion test, because it just makes more intuitive sense to me, and the second wave must have seemed like it just extra step, and it really uses information from the first derivative test. So and you can kind of see that there is a con cave up, shaped by just looking at how the function is behaving. So you don't even really need the second derivative test. I feel like it's an extra step. It's an extra derivative, but if it makes them to you on by all means, I whichever works best for you, I believe, um, so essentially the first true test, I feel like it's a faster way to do it, but if it makes it the second test, make sense to you, then use it.