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).

g(x) = x⁴ - 50x² on [-1, 5]

Accepted Answer

Below there today we're going to 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. Calculate the critical points for the function PF X is equal to 2 x 4 minus is 16 x 2 plus 32 on the interval. Closed interval -2.3 identify the absolute maximum and minimum values. Awesome. So it appears for this particular problem we're asked to identify the absolute maximum and minimum values. And we're also asked to calculate the critical points. So ultimately we're asked to solve for three separate answers. We're trying to find an answer for the critical points. That's our first answer. We're trying to find an answer for the absolute maximum and for the absolute minimum. So that's our last two answers. So now that we have an idea of what we're solving for now, let us start to solve for this problem first off by focusing our attention on solving for the critical points. So in order to find the critical points, we need to recall that we need to take the derivative of PFX. So taking the derivative of PF X, we will find that PX, where this prime simple indicates that we're taking the first derivative of this function of PF X, is going to be equal to. 8 X to the power of 3 minus 32 X. Fantastic. So now we need to set our first derivative of PX, so P X to 0. We need to set this equal to 0. So we will get that. 8 x 3 minus 32 X is equal to 0, which when we simplify this expression, we will find that it's going to be equal to x multiplied by x 2 minus 4 is equal to 0, which will give us answers of X equals 0, X equals 2, and X equals -2. Fantastic. So at X equals -2. P of -2. So if you plug in a value of -2 in 4 X, we will find that it's equal to 2 multiplied by -2 to the power of 4, because now we're plugging it back into our original expression. So 2 multiplied by -2 to the power of 4 minus 16 multiplied by -2 to the power of 2, plus 32. It's going to be all equal to when we plug that big old long expression into our calculator, it's going to be equal to 0. And at X equals 0 p of 0. So in an effort to save time, I'm not going to rewrite the whole expression again, but we are essentially taking our original function for p of X and we're plugging in a value of 0 in for X. So when we plug in a value of 0 in for X, we're going to get an answer of What drum roll, it's gonna also be equal to, it's gonna be equal to 32 because as you can see when we plug in a value for zero, the only number that's left is gonna be 32. So X equals 0, P of 0 is gonna be equal to 32. And for X equals 2, so P of 2 is going to be equal to 0, and at X equals 3. Once again, we're plugging in a value for 3 in for X, and when we do that, we will find that it's going to be equal to 50. Fantastic. So now, what we need to do is we need to evaluate the function at the endpoints and the critical points, so that we will get that P. So when you take P of -2, so when you take P-2 is equal to 0, P 0 is gonna be equal to 32, and P. Of 2 is going to be equal to 0, and P of 3 is going to be equal to 50. Fantastic. So that will mean, without a doubt, the absolute maximum value is going to be P of 3 is equal to 50, and the absolute minimum value is going to be 0 at X equals -2 and 2. So that will mean, without a doubt, our final answers all summarized together will be that our critical points. are going to be at X equals curly bracket of -2.0.2 curly bracket, and our p minimum, which once again the absolute minimum is going to be pin of -2 is equal to 0 and pin of 2 is equal to 0. And P max, which is our absolute maximum, so P max of 3 is going to be equal to 50, and that's it. We've officially solved for this problem. 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).
g(x) = x⁴ - 50x² on [-1, 5]

Key Concepts

Critical Points

Absolute Maximum and Minimum

Endpoints of an Interval

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