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