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Ind the absolute maximum and absolute minimum values of on the given interval_f(x) =Xe -X2/72 ,[-5, 12]absolute minimum valueabsolute maximum value...

Question

Ind the absolute maximum and absolute minimum values of on the given interval_f(x) =Xe -X2/72 ,[-5, 12]absolute minimum valueabsolute maximum value

ind the absolute maximum and absolute minimum values of on the given interval_ f(x) =Xe -X2/72 , [-5, 12] absolute minimum value absolute maximum value



Answers

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 12 + 4x - x^2 $, $ [0, 5] $

So we're wanting to establish absolute maximum and minimum values on the interval from 0 to 5. Um, so we're gonna restricted just within this. We see that the maximum is going to be located right here at 16. When X equals two and the minimum will be located right when access equal to five. So that's going to be seven.

Minimum values of function. Given it a go, the first thing we need to do is find the critical points and this critical points occur where the derivative of the function or the slope of the tangent line to the curve equals zero. So first thing we wanna differentiate dysfunction that comes out ast four minus two x. So now that we have the derivative, if we want to set that equal to zero and so for X Lipes when we find the X equals two. So this is one of our critical points. But we also have to consider the endpoints of Are given Interval so zero and five also are critical points. Now we need to figure out what the value of the function is that each of these critical points so richest plug each of these into original function. So this comes out is 12. Both thes terms are zero at X equals two. We find that the value of our Foshan is 16 and finally X equals five. We see our function gets a value that 25 seven no, no the the absolute maximum is the value of 16 and it occurs at X equals two and our absolute minimum is seven at X equals five

Okay, so? Well, here we have our function. F of X is equal to negative five. And then we're on the closed interval from negative one toe one. And that's to find the while the absolute maximum it. So if you think about this, I mean the function here f of X is equal to negative five. This is a constant function. This is just a horizontal line, right? So Okay, so we're gonna have a maximum men. Well, how we would go about solving this, Normally, it would take the derivative of our function. So we have our function. F of X is equal to negative five. While the derivative of prime of X, it's just equal to zero, right? The derivative of a constant is just zero. So are derivative is equal to zero. And then for, um, are critical values. We would say we would set the function equal to zero. So where is zero equal to zero? Well, everywhere. Right. So we could say that there are no critical. I mean, really, there are no click of values, or I guess you could say maybe that everything is critical value. Um, maybe since but there are no critical values. Um, So then what we do is just list the endpoints and say, Well, OK, eso over the given interval, right? You just evaluate the function at the two endpoints. But if you have value, if you evaluate ffx at any value, right, if you put the endpoints it if you do f of negative one whatever. Negative one, that's negative. Five. What is f of one? Well, that's five, right? What is what is left of any value on this interval or even not on the interval? Right. The value is I mean, negative five, right? The value is always negative. Five because of a constant function. So therefore, what is the absolute max? Well, negative five, right? The only valid stakes is negative. Five. So the absolute max is negative. Five. Um And what is that? Occur? Or it occurs on any value. So you could say, you know, negative one. Um, less than or equal to x less than or equal to one, right. Any value of X on the clothes interval. The value of the function is going to be negative. Five. So the absolute max isn't gonna find and the absolute men is also negative. Five. Right? Because we have a constant function. So, um so yeah, right. Um, don't over think that one, right. The value of dysfunction is always negative. Five. So therefore, the absolute max. Absolute men is negative. Five. That's the only rally dysfunction takes. All right, take care. Oh!

Hello and fix has given L N X. Okay. And the interview is gonna wanna come onto So we will do a chest X that is close to one by X. It's a critical point. Will win pity. And it is out of the in trouble, so we will not considerate. So we will find that Well, you were the endpoint. That is zero half of two. That is because to Ellen, to that is constant 20.693 So, Max, image half of two. That is born 693 So when the minimum is have have one, that is a question zero. This is the answer. I hope you're not sure. Thank you.


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