WEBVTT
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we will find the critical numbers of the function F
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of X equals X squared times Exponential of-3 X
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. So remember that C. In the domain of
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function is a critical number. If to uh any
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of two things happened. The first river activated point
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does not exist or it exists but he's zero.
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Mhm. So this is a definition of critical point
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function. So the first condition is at the point
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being the domain of function. And after that,
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if any of these two things happened then the number
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is critical number. You give a number of F
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. Of course. Okay, so here we find
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the first relative of F. In this case we
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have the product of two functions. So again,
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product rule is for derivative of the first factory square
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is two X times the exponential of native three X
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plus The first factor x squared times the derivative of
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eat the-3. X. is Each of the
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-3 x times the derivative of the exponents, which
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is-3. And we get to eggs eat the
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negative three eggs-3 x square feet and 93 eggs
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. And now we can take Common factor out X
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. E. to the-3 eggs. And we
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get inside parenthesis to minus three X. That is
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the first derivative of F. As expression eggs Eat
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the negative three eggs times 2-3 eggs. And
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this derivative here exist for any real number eggs.
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And we also know that the domain of this function
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is the real numbers. Because the formula defining F
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can be evaluated at any real number X. So
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we have the main of F equal the real numbers
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and the first derivative exists at any real number X
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. These two things together imply the stat. We
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apply that um The only critical numbers of F are
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those real numbers such that for which theory of the
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T. V. Zero? Because it cannot happen
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. The first possibility this one here because if conservative
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exists everywhere. So the critical points can only be
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Those values for is derivative zero. But now we
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start with this equation for conservative of f equals zero
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. That's the same. And saying that this formula
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here is zero and his ex he into the negative
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three X times two min minus three eggs equals zero
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. And this is equivalent. In fact that X
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equals zero or 2-3 eggs equals zero. That
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is because this factor here is not zero for any
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real number That is we have a product of three
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functions eggs times this exponential of-3 eggs. And
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these factor here to-3 eggs. Because the factor
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exponential, snap the three X can never be zero
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. It implies that the other two factors can be
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serious, any of them or both of them.
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That is or X equals zero, or the factor
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too many three X is equal zero. And that's
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the same. Are saying that X0 or eggs equal
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two thirds. If we solve this equation here for
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X and then these are the two values that for
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which the preservative is zero, that is. These
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are the critical point now and the critical numbers of
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so the only critical numbers of F R zero and
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two thirds. And that's the answer to this from
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.