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

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

Key Concepts

Critical Points

Critical points of a function are the x-values where the derivative is zero or undefined. They are important in determining where a function may have local minima, maxima, or points of inflection.

Differentiation

Differentiation is the process used to find the derivative of a function, which represents its rate of change. For polynomial functions, applying power rules yields a derivative that can be analyzed to find critical points.

Solving Polynomial Equations

Once the derivative is obtained, setting it equal to zero leads to a polynomial equation. Solving this equation, often by factoring, determines the x-values where the function's slope is zero, thus identifying its critical points.

Transcript

00:01 Let's find the critical values of this function, f of x equals x to the third minus 4a squared x, where a here, where a here is just some number that's not equal to zero.

00:14 So don't let that a throw you off.

00:15 It's not like another variable.

00:17 It's not like some.

00:19 A weird trick.

00:19 Just think of it as like being attached with that four, where this itself here is acting like its own whole number, not split up with, like i said, with another variable.

00:31 Okay, so for critical numbers, that's where the derivative is equal to zero or undefined.

00:37 So let's find the derivative.

00:39 So f prime of x.

00:41 This isn't that hard.

00:43 So then this becomes 3x squared, just a nice easy power rule.

00:47 Now, as i said before, this 4a squared is just acting like its own number.

00:52 So if we just think of it as like its own unit and then we just take the derivative, then the x just goes away.

00:59 So then this actually just becomes for a square like that.

01:05 And then, since we're finding critical values, you want to know when this is equal to zero or undefined.

01:10 It's not going to be undefined because there's no denominator or like square roots, so you're not going to divide by zero or have a negative under a square root or anything like that.

01:20 So we just have to find where it's equal to zero...

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