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So let's suppose see, FFC is an inflection point
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of our graph of our function of F. Of
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X. And so what we're gonna do is since
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we know that our point CFC is a inflection point
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, we can use the fact that the con cavity
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is going to change at X equals C. So
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this means than khan cavity changes at X is equal
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to see. And wherever the con cavity changes by
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the first derivative test, so can say by the
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first derivative test, our first derivative F prime of
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C will be a local extreme. Er that means
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it will be either a local max or minimum value
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for our um for our first derivative of death.
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And since our first derivative F has a local extreme
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A. At this point, C. We can
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say that by Fermat's theorem, which just says that
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a local minimum or maximum of a function F means
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that the derivative of that function is zero at that
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point. So if this F prime of C has
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a local extreme A. At that point, see
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that means the derivative of F prime, which should
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be F double prime C is equal to zero,
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so the derivative of F prime of C Is equal
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to zero. And this is just the second derivative
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of F is equal to zero at sea. So
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now we have proved that if CFC is an inflection
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point of fx, then that means that the con
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cavity changes at that point, which means that our
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first derivative is going to have a local extreme of
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their macs are men, which by Fairmont's thera means
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that our second derivative is going to be equal to
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zero.