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Problem 11 Medium Difficulty

(a) Find the intervals on which $ f $ is increasing or decreasing.
(b) Find the local maximum and minimum values of $ f $.
(c) Find the intervals of concavity and the inflection points.

$ f(x) = x^4 - 2x^2 + 3 $

Answer

(a) $f(x)=x^{4}-2 x^{2}+3 \Rightarrow f^{\prime}(x)=4 x^{3}-4 x=4 x\left(x^{2}-1\right)=4 x(x+1)(x-1)$
(b) $f$ changes from increasing to decreasing at $x=0$ and from decreasing to increasing at $x=-1$ and $x=1 .$ Thus,
$f(0)=3$ is a local maximum value and $f(\pm 1)=2$ are local minimum values.
(c) $f^{\prime \prime}(x)=12 x^{2}-4=12\left(x^{2}-\frac{1}{3}\right)=12(x+1 / \sqrt{3})(x-1 / \sqrt{3}) . \quad f^{\prime \prime}(x)>0 \Leftrightarrow x<-1 / \sqrt{3}$ or $x>1 / \sqrt{3}$ and
$f^{\prime \prime}(x)<0 \Leftrightarrow-1 / \sqrt{3}<x<1 / \sqrt{3} .$ Thus, $f$ is concave upward on $(-\infty,-\sqrt{3} / 3)$ and $(\sqrt{3} / 3, \infty)$ and concave
downward on $(-\sqrt{3} / 3, \sqrt{3} / 3) .$ There are inflection points at $\left(\pm \sqrt{3} / 3, \frac{22}{9}\right)$

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WB

William B.

October 21, 2019

Part 3 of the solution appears to conflict with the given answer in the textbook

Video Transcript

Okay, so we're being asked to find the intervals in which ever is increasing and decreasing the local maxim men and intervals of Khun Cabinet and inflection point of FX. So in order to find where efforts increasing decreasing, we apply the first derivative cast. So we have to first take the derivative of ethnic so proud of X is equal to for X cube miners for X Oh, and then that could be further simplified by. And we said this equal to zero, but first going to simplify this further so we can actually pull out for eggs. And now leave us with X squared minus one. And we said the sick or a zero and we saw for different cases. And Joe So we have for X equals, you know, and explorer and minus one equal zero. Well, this cages and excess equipment here And since we are, add the one over and take the square root. This cloud of one is one. But remember that when you take a square root, you have plus and minus c. Yeah, negative one and one. Or you could also look at this as X plus one times X minus one no and then solved after a zero. And I'll give you the same answer. Um, so now that you have these values, I like to make a sign chart, as I've explained before, So I will. I have thought this would be the number line right here. This is horrible line. This is the number line, and I will put values on which I'm interested in, which, which are these numbers. And then on my ex side, I put the fact that the multiplication, which I'm interested in justice in this case, will be for X and X square minus one and desirable parts of the function serving multiplied to give us a prime. And then we're valuing. We're gonna look at negative one. It's a restaurant zero and one, and this over. And so this is basically like a sign chart. And then when you multiply these together, you get the sign of a prime. So anybody less than negative one for club in value, less negative, one for four executed negative number, and then then for any value between negative one and zero. You also still get a negative number. Anybody read them to you on one to get, probably remember and a positive number. And then for X square minus one. People haven't body's less than negative. One used to get a positive number because you're squaring anybody. The twenty one negative one and zero you got a negative number and then negative number of Quinn June one and then a positive number. So now you do. Now you multiply across the positive negative, negative, negative negative is positive projects have negative, negative and positive time father's posit because now that this tells us the Stynes of prime we know the intervals on wish their increasing and decreasing, which I'll right here. So it is increasing where have primacy Positive. So it is positive between negative one and zero. So to make it a horn Joe union and it's also positive from one to infinity. And then it is decreasing where it is negative. So it'LL be negative. Infinity too negative one and then the other one. And we can also find out a little call Max and men. So write our max. Remember that D's tells us the direction this tells us the direction with the front graphic milling So this decreasing, increasing, decreasing and include and increasing look. So where are function is going from negative two positive. We have a local men so our men and on the opposite is true for itself is going to be positive to negative. You have a local Mac. So then this occurs our deal and negative one and the local men's Akos That negative one and one for the reasons stated. Now to figure out the front cavity, we have to take the second derivative so and said that equal to zero. So the second derivative of this is twelve x, where Barnett's whore This will be twelve x squared minus for and he said the secret deal. But we can also simplify this for them can pull out of four. He will give us three x squared minus warm and we said that and resolve for three extra minus one and I'LL give us X is equal to possum minus quiver of one over three And now we are going to repeat a signed shirt again. But this time for Akane Cavity. So this time it will be We'LL have negative one of three I mean a positive one over three and then our only time that we're interested in is direct squared minus one. And if you plug in values less than negative on over three, you get numbers that are positive. You plug the numbers between these negative with one of the three in positive wanted to refigure numbers that are negative. And then if you plug numbers greater than we wanted through, you get positive. So if and these tells us a sign of a crime of double and so f promise positive, which it is in this case, it is. Khan came up that negative has can't give down and it was covered with positive. It's conking up. So our intervals so con right it is is negative. Infinity, Teo Negative one over three and union Ah, positive. One of the three to infinity. Maybe this should be yeah. And then down conquered down occured between negative one of three two positive one of the three and they're inflection Point occurs when we have changes in the signs of con cavity and dis occurs at negative one of a growing problem in its race. The inflection point point it occurs that plus a miner's one over three on DH. That is all