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Problem 55 Hard Difficulty

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

$ f(t) = t - \sqrt[3]{x} $, $ [-1, 4] $

Answer

Absolute minimum value $\left( \frac{1}{3} \right)^{\frac{3}{2}} - \left( \frac{1}{3} \right)^{\frac{1}{2}}\approx -0.3849$ which occurs at $t=\left( \frac{1}{3} \right)^{\frac{3}{2}} \approx 0.19245$ ;
Absolute maximum value $4- \left( 4 \right)^{\frac{1}{3}} \approx 2.412598948$ which occurs at $t=4$

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Video Transcript

let's find the absolute minimum and absolute maximum values as a function F. F. T. Will t minus cubic root of tea on the closed interval from negative 1 to 4 first. We know that F. Is continuous on that close interval. And for that reason we know it attains its extreme values over that interval. And we know more than that that the extreme bodies are attained either at the end points of the interval or at critical numbers of F. So we gotta find the critical numbers numbers of F. For that to calculate first the first solar derivative of F. And that's one minus uh the relative of cubic root of T. Yes. If we write that as T. To the one third is one third times T to the one third minus one And that is 1 -1 3rd times t. to the -2 certs. Yeah. Yeah, He did the negative 2/3 in that's equal 21 minus 1/3 times T. To the two thirds. And that's discretion for the first derivative of F. And this derivative At zero does not exist. It's not defined there. Now Syria is a point in the domain of F. in fact effort 0, 0. And because Syria is in the domain of the function and the relative of F does not exist exist at that point. We can say that uh equals zero is a critical number of earth by definition. Yeah. Now we got to solve the equations after a difficult zero and that's equivalent to one minus 1/3 T. To the two thirds equal to zero. And that's the same as three T to the 2 3rd Equal one. And in fact this is the same as T to the 2/3 equal 1/3. In fact we are we got to consider all the way through this equivalence is here, we got to consider T different from zero in order to write this expression like this. Okay, so we know that this is the case because the consider was already treated separately here above where we say that there is no derivative there and for that reason and the point being in the domain we know that that value of equal serious a critical number of deaths. We have this equation here, we can say that this is the same to T square equal one third To the to the 3rd power. And here we have two solutions to the equal, more or less, one third to the third. The square root of that of course let's say. And that is more or less, we can write this as one third to the three halves and we have two values and the two values here let's say this about in order to have an idea of zero point. Um See your point so and 19 245, That's T to the 1/3, That's 1/3 to the three hats. And both fats are in the interval from 21 to 4. So both of them had to be considered. So we have three critical points. Has three critical numbers In 91 4. Mhm. Yeah. Which are t equal -1 3rd to the three House T equals zero and t equal one third to the three house. That's the three critical numbers of F in the interim negative 14. And now we um Got to evaluate the function at those critical numbers and at the end points of the interval -14. And we get the following first at the endpoints effort negative one is equal to negative one minus Nettie 1- Cuba group of native warn It is Native one Cuba group of 91 is 91 And with his minus here in front of the cubic root is Plus one. That is serious. Okay, so now F at four, which is the writing point of the interval is four minus cubic root four. And this is about in order to have an idea of the value two point 41 259 8948. And then we evil wait three critical numbers F at negative one third to the three house, this one here. Mhm. Yes, negative one third to the three house minus one third to the three house to the one third because the cubic root of the number is the same as Putting into the power of 1/3 and this is the same as negative one third to the three house plus. Because here I made I forget the sign. Sorry here it's negative here. Inside here in front of the branches. Is here is this negative of the formula of the function. Okay. And then we have the cubic root that is racing to the one third. The number where we are evaluating which is negative one third to the three house of. I forget this negative sign here. Well. And now because cable group we have the result is negative with this negative here out, his plus is a plus sign result get plus one third And the power end up being 1/2. Yeah. Okay so this amount this number here, Yes mm 0 0.38 uh 49. And now f at zero which is the other critical number is zero. And if at 1/3 two D three house is one third to the three house minus the cubic root. That is raised to the one third. The number where we are evaluating the function that is 1/3 to the three House and that becomes 1 3rd To the three house -1 3rd to the One House. And that's about zero 3849. So from this Values here, from these five values, we got to find the maximum and the minimum values and functions we know that these are the maximum value is 2.41. Uh The Writing .4. Yeah and the minimum value is yeah, This one here negative 0.3849. About that at 1/3 to the three house. So with that we can say the answer to the problem. Mhm. Then the absolute minimum value of F in the interval from 91-4 is um 2.41 and four is 4 cubic root of four which is about that's the exact value. But to have an idea of what value it is, it's about Uh 2.41 2598. To find an 8948 in that absolute minimum value. Sorry, I make stops or I may stop the answers here because I'm talking about the minimum, I wanted to say. Um The minimum value, we set this one here. So I made a mistake here. Sorry. The minimum value of the function is negative. Zero point is one third of the three house, 1/3 to the three House -1 3rd to the one house, which is about which is what -0.3849. That is correct. Which of course At 1/3 to the three health, which is in fact one of the critical numbers at the critical number To equal 1/3 to the three house. And this in fact is about 0.19 245. And they are now the absolute minimum maximum value of F In the interval 94, 1 is now it's a value. It said before is the maximum 2.41. That is four minus cubic group before is four minus cubic group of four, which is about 2.141 259 8948. And that value of course at uh the writing point of the interval to equal for so, Mhm. Mhm. Your final answer, which is this one here? This sentence here says that we have a natural max actual minimum value of the function of one third to the three house minus one third to one half. And that's about 19 0.3849. And that of course at the critical number 133 House, which is about 0.19245. And the answer of maximum value of half in the close interval is four minus cuba group four, which is about two point 41259484848948. And that of course at the right end point of the interval t equals four. So we have the exact values, extreme violence of the function and the except values in the interval negative one for where they occur

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