Problem 30 Medium Difficulty

Find the critical numbers of the function.

$ f(x) = x^3 + 6x^2 - 15x $

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Mathrix P.

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Madi S.

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Chris T.

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Video Transcript

we're going to find the critical numbers of the function F of x equals x cubed plus six x square minus 15 X. So we know that the numbers into the main of the function. Search that derivative at that point doesn't exist or you exist And is equal to zero are called critical numbers. So this is our definition of critical number points in the domain of the function where the everybody either does not exist or exist any sequel to syria. So the first thing we got to say here is the main of the given function. Talking about this now is the real numbers because it's a polynomial well defined at any real number. That's one thing. The other thing is that the derivative of F at any point Is given by three eggs square Plus 12 Eggs -15. And this derivative yes, define or exist. Uh huh. Every real number. So for this reason we are not going to look numbers in the main where the derivative does not exist because everybody exists at all real numbers. Then we got to look for numbers where numbers in the domain of the function that is in the real numbers where the derivative is Syria. So given that the first derivative exists and every your number. And we have to find okay, where the relative peace. He's uh every Where the derivative or f derivative is zero because we are not going to find points. Worries that everybody doesn't exist because it exists everywhere. Yeah. So you're looking for points in the domain that is in the real numbers were activist Cyril. And for that really Equated first serve that to zero. That is three eggs square Plus 12 X Men 15 0. And that's the same as three times Factory. Now it's number three Out of the polynomial. So we get three x square. That's four eggs minus five. It means that for the first conservative to be called zero, sufficient to look at the series of this phenomenal here there is a factor three is not changing that. So this is going to X square plus For eggs -5 equals zero. And now what we can do here, we could obviously apply the formula for the roots of a polynomial of degree two. But we can factor it out because in this case is easy. You find two numbers whose product is -5 and whose sum is four. Those numbers are five and negative one, five times negative one is negative five And five plus 91 is 5 -1, which is four. So this is factoring Factory Ization is these pollyanna here, we can verify that easily. If we distribute. Make the distributive property of multiplication with some and different so and obstruction. And we can verify this, correct. And so now we have factor at the polynomial. We can see clearly the two routes of the polynomial got to be five negative fights or X. is negative five or eggs equal one. And then this is terrified that evaluated at one or a negative five, we got snow directive. So the critical numbers of the function F equal X cubed plus six X square minus 15 eggs. Are I got to five and why? Yeah. And these are the only critical points of this function because they are the only drill numbers that nullify the first derivative and there are no values of X for which the first derivative does not exist because it is well defined and exist everywhere that is in they set of real numbers.