Find the critical points of the following functions on the giv...

Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the critical points of H of Z is equal to 5 of 2/3 minus Z 5/3 on the interval square 0.22. What are the absolute maximum and minimum values of H of Z? Fantastic. So it appears for this particular problem we're asked to solve for three separate answers. Our first answers we're asked to determine what the critical points of this function of H of Z is, and our last two answers we're asked to solve for is we're asked to. Figure out what the absolute maximum and minimum values of H of Z are. So now that we know what we're ultimately trying to solve for for three separate answers for the critical points and for the absolute maximum and minimum values for this function of H of Z. In order to solve this problem, first off, let us start by solving for the critical points. And in order to solve for the critical points, as we should recall, we must first find the derivative of H of Z. So let us know that the derivative of H of Z is going to be H of Z, where this prime symbol denotes the first derivative of this function of H of Z. So H Z is equal to 10 divided by 3. Z, so it's gonna be 10 divided by 3 multiplied by Z to the power of, so we need to take Z to the power of -13 minus 5 divided by 3, multiplied by Z to the power of 2/3. Fantastic. And now we need to set this equation of H Z equal to 0. So when we set it equal to 0, we will find That it's equal to 10 divided by 3 z to the power of negative 13 minus 5 divided by 3, and we need to take 5 divided by 3, and we need to multiply it by Z to the power of 2/3, and this will be equal to 0, and this will simplify to 10. Minus 5 Z divided by 3 multiplied by Z to the power of 1/3 is equal to 0. And when we solve for Z, we will find that Z is going to be equal to 0.2, so it's going to be equal to 0 and 2. So when Z is equal to 0, let us know that when we plug in a value of 0 in for Z, we will get that H of 0 is equal to 5 multiplied by 0 to the power of 2. 3rds minus 0 5/3 is equal to 0, and when Z is equal to 2, H of 2 is going to be equal to 5 multiplied by 2 of 2/3 minus 22 to the power of 5/3, which is going to be equal to, when we plug that. So, actually, before we start calculating this, or rather, when we calculate this, when we plug this into our calculator. We will find that it's going to be ultimately equal to 4.7623, when we round to 4 decimal places, but if it's easier for you to solve, you can actually calculate. 2 to the power of 2/3 and 2 of 5/3 separately and then perform the subtraction. It's whatever's easier for you, but if you have a graphing calculator, you could just easily plug in this expression into your calculator. Just make sure you have your parentheses, or else you'll get incorrect. So you need to put the power in parentheses. In your calculator to make sure you get the proper answer. So when you evaluate. H of 2, you should get 4.7623 when you round to 4 decimal places as an answer. So now when we evaluate H of Z at the end points and the critical points, we will find that H of 0 is equal to 0, and H of 2 is approximately equal to 4.8 when we round to one decimal place. So therefore, there will be an absolute maximum at 4.8 at Z equals 2, and an absolute minimum will be 0 at Z equals 0. So that will mean as a summary of our final answers, the critical points are going to be at Z equals curly bracket, so we need to say Z is equal to curly bracket of. 0.2. Don't forget your other curly brackets. So 0.2 curly bracket. Awesome. So those are our critical points and then our absolute maximum value, which we'll call H max of zero, is gonna be equal to 0. And finally, our H min, which is our absolute minimum value of 2, is going to be approximately equal to 4.8. And that's it. Those are our final three answers. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]

Key Concepts

Critical Points

Absolute Maximum and Minimum

Evaluating Functions on Intervals

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