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Consider the following function f of x is equal to x times x minus 8 to the power of 3.
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We want to find this function's points of inflection.
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Points of inflection correspond to values of x for which the second derivative of f is equal to 0.
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So we're going to need to compute the first and second derivative of f.
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Let's start with the first derivative f prime of x.
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To find f prime of x, we're going to utilize product rule, and we find x minus 8 to the power of 3 plus 3 times x times x minus 8 to the power of 2.
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Next, we want to find f prime prime of x.
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So we're going to differentiate f prime of x again, and our first term here will give us 3 times x minus 8 to the power of 2.
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And our second term, utilizing product rule, again we find 3 times x minus 8 to the power of 2 plus 6 times x times x minus 8.
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So now we can simplify this expression, and we find 6 times x minus 8 squared plus 6x times x minus 8.
01:45
And now we can factor out a 6 times x minus 8, and we obtain 6 times x minus 8 times x minus 8 plus x.
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Simplifying to 12 times x minus 8 times x minus 4.
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So this is our simplified form for f prime prime.
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So now to solve for the points of inflection, we need to find values of x for which f prime of x, f prime prime of x, is equal to 0, which will be the case when our polynomial 12 times x minus 8 times x minus 4 is equal to 0.
02:40
So here we have two points of inflection, x equal to 4 and x equal to 8...