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Yeah. We want to find the critical numbers of
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the function F of X equal to x cubed plus
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X squared plus two X. So, first we
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find the derivative of F. That function is six
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X squared plus two eggs plus two. And this
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is relative exist at any ex any real number eggs
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. The main in fact of the function is the
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whole real numbers. Because this expression is paranormal.
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Is it fine for every real number X. So
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the men of F. S are the real numbers
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and in a relative exists for every eggs in the
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real numbers. That is is two facts together implies
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that the only critical points of function F Are those
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eggs for which there are TV zero. Mhm.
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Yeah. That is because remember the definition of critical
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number of a function. Yeah. All right.
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The X. Or values in the domain of the
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function for which the relative does not exist or It
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exists and is equal to zero. So uh that's
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why we look first at the domain, the function
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that is the real numbers. And then we see
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where derivative exists. In this case it exists for
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every real number. That is for every X.
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In the domain of the function. Therefore the only
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critical points or numbers of these functions are those values
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of X. In the real numbers for which the
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derivative is equal to see. Mhm. So the
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equation we gotta solve is this f derivative equal zero
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. And that's equivalent to looking at the formula here
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, derivative six X square last two eggs Last two
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equals 0. And that's the same thing in two
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common factory at two times three X square plus eggs
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plus one. Because two is against a constant different
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from zero is equivalent to, sorry, here is
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equal zero. And is going to that the expression
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inside parenthesis through three X squared plus X plus one
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Be equal to zero. That that is the equation
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we get us solved with this one here because sold
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in this equation we have all the eggs for which
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the first derivative of the function is equal to zero
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. So let's find that by applying the formula.
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Or best, we're going to calculate the discriminate of
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this equation because second degree polynomial equals zero. So
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you see it's criminal expression which is the square minus
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four K. C or B is a division of
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X to the one a suggestion of X square and
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sees an independent term. So in this case is
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equal to one square Wayne is four times 3 times
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one That is one 12. That is-10.
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And because this expression is going to appear inside the
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square root in the general formula, dissolution of the
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second order and a question, you know that with
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this condition here, we won't have real solutions,
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then, um There is no number X. And
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the real numbers such that yeah, three X square
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plus six plus one equals zero. That is the
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same as saying that the first derivative of F Different
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from zero for all eggs, Nah, real numbers
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. And then taking into account what we said above
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, we can conclude that this function has no critical
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points and that the final answer cause this problem.