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Find the absolute maxinum and minitum values of the function f(z) the interval (0.2. Note: Caleulator ansWcrs are not acceptable. You must give Calculus explanation...

Question

Find the absolute maxinum and minitum values of the function f(z) the interval (0.2. Note: Caleulator ansWcrs are not acceptable. You must give Calculus explanation, follou: the stepr outlined closs:Step Compute f' (€) and find the critical numbers Step 2. Fiud the values of f at f(c) , us well as the values of f atthe endpointa and i.e. f(a) and f(b). Whichever One of the above values the largest= that will turn out to be the Step 3 and whichever one of the above valucs is the smnallest

Find the absolute maxinum and minitum values of the function f(z) the interval (0.2. Note: Caleulator ansWcrs are not acceptable. You must give Calculus explanation, follou: the stepr outlined closs: Step Compute f' (€) and find the critical numbers Step 2. Fiud the values of f at f(c) , us well as the values of f atthe endpointa and i.e. f(a) and f(b). Whichever One of the above values the largest= that will turn out to be the Step 3 and whichever one of the above valucs is the smnallest , that will absolute maximum value FTE out to be the absolute mninimum value



Answers

Find the absolute maximum and minimum values of $f$ on the given closed interval, and state where those values occur. $f(x)=\left(x^{2}+x\right)^{2 / 3} ;[-2,3]$

But the thing you probably want to find the absolute maximum and absolute minimum values on the given interval. So we're going to consider ffx Equalling need to the negative acts And it's either the -2 x Very like from 0 to 1. So consider after zero one. But then we also want to consider critical points in finding the absolute maximum and minimum. So with that we're going to consider after crime effects and we're going to have f prime of X. Um we get a zero right here 0.693. So plugging that value in. Mhm. You see that this is going to be the absolute maximum at 0.25 And the absolute minimum is up zero.

Here. We have to function after box is equal to one minus X to the two thirds power on the closed in developed from negative 8 to 8. So we first take the derivative of our function So f prime off X while we take throughout the term. By term, we have a fractional exponents. You okay? Right. So the derivative here is just Well, we just get to third X to the two thirds minus one. That's two thirds minus three therapists. Negative. One third. Right. Um, so, aan den times when you have a native born here in front, therefore the derivative of Prime of X is gonna be equal to negative two thirds X to the negative one third. Okay, so there's a derivative now find the critical values. We just take a negative, and we set it equal to zero. So if we do this, Well, this was just imply, right? We could if we want to mean much, play both sides By what native three halves that's gonna cancel out the coefficient on the X, but still make it zero. All right. And then we have actually one third, um, equals zero. That that's still that implies that X is equal to zero. So this may look a little scary, but it's really not because we just get here. That X is equal to zero. Those are critical value. Okay, so we list out all critical values and endpoints. So we have a negative eight is the one end point, then the critical value of zero and the other end point of eight. Now we evaluate our original function at those values. So let's do F zero first. So f zero is equal. Talk about one minus zero, which is equal to one. So f zero is equal toe one and then do f of negative eight. So f off. Negative. Eight. Well, that's equal to one minus negative. Eight to the two thirds. Um, okay. The cube root of negative eight, which is, um, two and a few square it. Get four. This is what you just get. One minus four, which is native three. So f of negative eight is negative. Three and f of eight. Well, we get one minus 8 to 2 thirds. So again, 8 to 2 thirds is the cube root of eight squared. That's four. So, again, we get one minus fork. So again we get half of eight is also equal to negative three. So not to find the absolute max and men. So over the interval, native 8 to 8 or the absolute maximum is the largest output. And we see that is the only positive value, which is one right. So the absolute max is equal to one, and it occurs when X is equal to zero. So that X equal to zero now for the absolute men. Yeah, well, that's the smallest output is negative. Three, right? We absolutely men is negative. Three. But what is it occurred? Where the coach at two values right that occurs at X, being equal to negative eight. Okay. And also right and also x be equal fate.

All right. So for this function we need to find some critical values. So we're gonna find the derivative bring forward down in front bring three down. Sorry, that is a two on this problem. Not a three And the two down in front And then the River 30. More, silly. Zero. And solve we're gonna take out four X x squared minus one means we have critical values at 01 and negative one. They're all in our interval. And so we're going to evaluate all of them. So what's F of zero? Zero minus zero plus three? Which is three? What is F of 1? That's 1 -2 plus three. Which is to The F of -1. That's 1 -2 plus three. Which is to and then we got to check our endpoints. What is f of negative to -2 is 16 minus eight plus three which is a weapon. And once every three We plug three in there, We end up with 81 minus 18 plus three. So we end up with 81 -15 which is 66. And so f of three is going to be our absolute max for sure. And then we have to absolute minimum we have a tie. So that's the answer for that problem.

Let's find the absolute maximum and minimum values of the given function. So it's going to be half of ax equals X over X squared What's one? It's gonna be from 0 to 2 quickly. So sentiment, we want to consider the derivative crime affects, we end up getting this. So once we take the derivative through the collection role, we can now set equal to zero and we find that equals zero and X equals one. So then what we can do is plug in F of one, just one half that won't pick something to the left of that. So we'll pick F of zero. You get zero and then of yep, Get your .4. So we can see based on this, is that the slope or that the uh we want to look at a crime of one. Actually we see the um we'll go f prime of 1.5 for example, uh 0.5. We see the slope is positive and it reaches zero, then it goes negative. So with that in mind, we want to consider the function. We treat you right here. So that's your final answer.


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