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If π of π₯ is equal to ππ₯ cubed plus seven π₯ squared minus eight π₯ plus nine and the second derivative where π₯ is equal to nine is equal to negative nine, find π.
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Well, in order to solve this problem, what weβre gonna need to do is actually find the first and second derivatives of our function.
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So in order to actually find the first derivative, what weβre gonna do is actually differentiate each term individually.
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So our first term is gonna be three ππ₯ to the power of two or three ππ₯ squared.
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And just to remind us what we did to get that, so if weβre differentiating, what we do is multiply the coefficient by the exponent, so π multiplied by three.
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And then, weβve got π₯ and then to the power of and we actually reduce the exponent by one, so three minus one, which gives us a three ππ₯ squared.
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So okay, great, what weβre gonna do is use this to actually differentiate the other terms.
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So then, weβre gonna get plus 14π₯ minus eight.
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And thatβs because actually if you differentiate positive nine, so just an integer on its own, you get zero.
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So thatβs our first derivative.
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All we need to do now is find our second derivative.
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Well, in order to find our second derivative, all we do is differentiate our first derivative which is gonna give us six ππ₯ plus 14.
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Okay, great, so now, we got our first derivative and our second derivative.
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Well, we know that our second derivative when you actually have π₯ is equal to nine is gonna be equal to negative nine.
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So letβs use this to actually solve our problem and find π.
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So therefore, we can say that negative nine is gonna be equal to six π multiplied by nine β and thatβs because weβve substituted in nine for our value of π₯ β then plus 14.
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So weβre gonna get negative nine equals 54π plus 14.
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So therefore, if we actually subtract 14 from each side of our equation, we get negative 23 equals 54π.
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So then if we actually divide each side of our equation by 54, we get negative 23 over 54 is equal to π.
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So therefore, we can say that if π of π₯ is equal to ππ₯ cubed plus seven π₯ squared minus eight π₯ plus nine and the second derivative when π₯ equals nine is equal to negative nine, then π is gonna be equal to negative 23 over 54.