Locating critical points Find the critical points of the follo...

Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(t) = 1/5 t⁵ - a⁴t

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the X coordinate of the critical point of the function F of X equals a quarter of X minus b cubed X where B is a constant and B is not equal to 0. A says X equals B cubed. B says it's B squad, C says it's B, and D says it's negative B. Now, if we're going to determine the X coordinate of the critical point of the function, let's ask ourselves, what do we know about critical points? Well, recall, OK, that a critical point. Critical point. Under function F of X. Or sorry, off F of X. Is any point, OK? X equals C in the domain. That satisfies either of the following two conditions. Where first, either the first derivative. OK, of our function equals 0. Or Second, OK, where our first derivative is undefined, OK. Where our first derivative is undefined and FF X is still defined. And now, if we can figure out or if we can get a value of X within the domain that satisfies either of these two conditions, that means a critical point exists uh at, sorry, a critical point will exist. At those X coordinates, OK? So, let's go ahead and test both of these conditions. But before we do that, of course, it means that we'll need the first derivative of our function. So, let's go ahead and find that. So now, we need to differentiate FFX. Now, here, if we were to differentiate F of X, OK, then the first derivative of X is going to be equal to 1/4. Well, we can differentiate 1/4 of X to the 4th using the power rule. So that's gonna be 4 multiplied by. 1/4 to the pore of X. 1/4 X to the pore of 4 minus 1 minus b cubed by x to the 1 minus 1. Again, we can apply the po rule here and in this case, this will simplify the X cubed minus b cubed. OK, so this is the first derivative of F of X. Now, let's test both of the conditions first. Starting with our first derivative being equal to 0, OK. When F of X, X, OK, equals 0, then we're gonna get X cubed minus B cubed to be 0. That means x cubed will be equal to b cubed, which must mean since the powers are the same then the bases are the same. In other words, X must be equal to b. So in this case this tells us when F X equals 0, the X coordinate of the critical point is B. What about when it's undefined? Well, if we think about FX, OK. Notice, if you think about the nature of FX, it's X cubed minus B cubed. This is a polynomial and because it's a polynomial, it is defined for all real numbers. In other words, this is not possible. F X cannot be cannot be undefined because it is a polynomial, and there are no additional critical points from undefined values. Therefore, that tells us that the X coordinate of the critical point based on our first condition is going to be B. Thanks a lot for watching everyone. I hope this video helped.

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(t) = 1/5 t⁵ - a⁴t

Key Concepts

Critical Points

Derivative

Polynomial Functions

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