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If 𝑓(𝑥) = 𝑎𝑥³ + 7𝑥² − 8𝑥 + 9, and 𝑓″(9) = −9, find 𝑎.
If 𝑓 of 𝑥 is equal to 𝑎𝑥 cubed plus seven 𝑥 squared minus eight 𝑥 plus nine and the second derivative where 𝑥 is equal to nine is equal to negative nine, find 𝑎.
Well, in order to solve this problem, what we’re gonna need to do is actually find the first and second derivatives of our function. So in order to actually find the first derivative, what we’re gonna do is actually differentiate each term individually.
So our first term is gonna be three 𝑎𝑥 to the power of two or three 𝑎𝑥 squared. And just to remind us what we did to get that, so if we’re differentiating, what we do is multiply the coefficient by the exponent, so 𝑎 multiplied by three. And then, we’ve got 𝑥 and then to the power of and we actually reduce the exponent by one, so three minus one, which gives us a three 𝑎𝑥 squared.
So okay, great, what we’re gonna do is use this to actually differentiate the other terms. So then, we’re gonna get plus 14𝑥 minus eight. And that’s because actually if you differentiate positive nine, so just an integer on its own, you get zero. So that’s our first derivative.
All we need to do now is find our second derivative. Well, in order to find our second derivative, all we do is differentiate our first derivative which is gonna give us six 𝑎𝑥 plus 14. Okay, great, so now, we got our first derivative and our second derivative.
Well, we know that our second derivative when you actually have 𝑥 is equal to nine is gonna be equal to negative nine. So let’s use this to actually solve our problem and find 𝑎. So therefore, we can say that negative nine is gonna be equal to six 𝑎 multiplied by nine — and that’s because we’ve substituted in nine for our value of 𝑥 — then plus 14. So we’re gonna get negative nine equals 54𝑎 plus 14.
So therefore, if we actually subtract 14 from each side of our equation, we get negative 23 equals 54𝑎. So then if we actually divide each side of our equation by 54, we get negative 23 over 54 is equal to 𝑎.
So therefore, we can say that if 𝑓 of 𝑥 is equal to 𝑎𝑥 cubed plus seven 𝑥 squared minus eight 𝑥 plus nine and the second derivative when 𝑥 equals nine is equal to negative nine, then 𝑎 is gonna be equal to negative 23 over 54.