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$3-4$ Use the level curves in the figure to predict the location of the critical points of $f$ and whether $f$ has a saddle point or a local maximum or minimum at e...

Question

$3-4$ Use the level curves in the figure to predict the location of the critical points of $f$ and whether $f$ has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions.$f(x, y)=3 x-x^{3}-2 y^{2}+y^{4}$

$3-4$ Use the level curves in the figure to predict the location of the critical points of $f$ and whether $f$ has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions. $f(x, y)=3 x-x^{3}-2 y^{2}+y^{4}$



Answers

$3-4$ Use the level curves in the figure to predict the location of the critical points of $f$ and whether $f$ has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions.
$f(x, y)=3 x-x^{3}-2 y^{2}+y^{4}$

For the given problem, we want to use level curves in the figure to predict the location of the critical points. Um So then we can also use the second derivatives tests to confirm this. So what we can do is we can take the partial derivative of F with respect to X. So we end up getting three X squared and then minus three X. Or minus 33 Y. And then if we set this equal to zero, um we'll end up getting that, there's going to be a saddle point At 00. We can also take the partial derivative with respect to Y and get three White Cube minus three X. And again we want to set that equal to zero To find the partial derivative and we see once again 00 that subtle point. And then we also get a local minimum at 11 set their final answer.

The problem is used, the level curves in the figure to predict the location of the critic points off us and the y hav e Stata Center point or local maximum. How anyone at issue quickly poor point. I explained our reasoning. Then use this magnitude with your past to confirm your predictions. First, let's look at this graph. So it is our wares at other point. Next one one next one. Next one, a fax. Why half the local minimal? And that's the point one zero a fax. Why Haas? Local Maximum. That is a point next to one zero one one one next to one facts. Why have the Seidl points? The reason is the tickets of one one. For example, if you look at is the work you lying here, we can see this point. It is no Cole minimal, but they feel look at it's a horizontal line here. This is a local maximum, so shouldn't be a critical point. But it is neither a local minimum. No local maximums. It is standpoint. It's here. Nick, Do you want one next one Next one local minimum. Why zeroes? It's a local maximum. Haven't you? Next two one zero one one one Next to one recital point, you re compute. Thanks. This is three months three X koi. Why seek or two negative, or why? Minus a plus for Wise Cube. Why is he going to zero XX? It's like a negative six sects. Well, why? It's the connective or US twelve y scores. That's the point. Make two one one. We have x x. The point if you want half XX is able to six off. Why? Why if they caught you? I ain't So is cultural forty eight zero f x x greatest and zero. So this is a local minimum. I disappointed. Nick wanted to one. So a fact Sex is also six have explained that my wife's also it. But these are the eight is sicker than cereal. A fax fax is security and Cyril So this's house Our local minimum point one zero x x is the connective six. This is last time. Zero Ask why Why is he going to connect your for the is able to twenty or risk with zero Toby Half this is a local maximum. At this point. Next one zero off Axe axe is a six one. Why is sick or so we have? It is equal to negative twenty four. So last San Siro, This's a saddle point. Our first tuition notice that if we liked half Axle is equal to zero. Why is he with zero? Can Self X is equal to plus or minus one? Why is he with Cyril so minus one all the six points? A critical point? That is a point to one one. So we have X X is equal to negative six. Why lie? It's a photo. So is he put negative forty eight so small and zero notices a subtle point one two one something. A fax fax is a negative six wise. They tow it is the connective forty eight. It's less than zero, since this's how saw a subtle point.

Ants clears the when you read here. So if we look at so level currents near 11 one, I'm gonna write it down here. So 11 one negative one and negative 10 They look like hyper Ballis. So this tells us that when we move away from the origin, the value of the function, it goes down or some directions and that increases in other directions. So they're saddle points. So looking at the contras around the negative 11 and negative one negative one negative 11 negative one negative one. If we go away from these values, move away, then the leaves curves increase in volume. So the point negative 11 and negative one negative one are minimal Good. Here and then up a sp for a saddle point for short and around 1110 You see that as we move away from the point, believe curves, they decrease. So it has to be a point of maximum for the second part. F x combo Why is equal to three x minus X cubed furnace to square plus y to the fourth f x ex con? No. Why? First is negative for Why less for why It's you of double eggs, x comma y when we get the jury. But I've begun negative floor must. Well, why square so from the second derivative test, we don't 24 acts minus some need to. That's I swear. Then putting f X equal, Cyril, we get excess equal to one common one or negative one. Inputting y equals zero you get why is equal to zero negative 11 So we're gonna do the second derivative test of these points and the critical values we get our 10 when one one negative one negative, 10 negative 11 and negative one negative. And it tells us after we do the second derivative test that the points 11 11 one negative one and negative 11 or saddle points and 10 is local mats and negative 11 negative one negative. One firm

The base of the first thing we're gonna do is we're gonna find F sub X is equal to three X squared minus y and F supplies equal to three y squared minus acts. So setting this is equal to zero Be enough with wise equal to three x squared. Similarly saying three us crazy with Xs equal to zero and substituting in why we end up with 27 extra. The power of four minus X is equal to zero. Giving us X is equal to zero and accidents equal to 1/3. Therefore, we end up with a corresponding wise equal to zero and why is also equal to 1/3? So we have zero comma zero and 1/3 comma 1/3 as our two points Using the discriminate, we're gonna end up zero comma zero is a saddle point and 1/3 comma 1/3 is a local minimum


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