Advertising for a certain brand of widgets is terminated and sales begin to drop. In fact, t weeks after the ads stop, sales are f(t) cases, where f(t) = 1000(t+8)^-1 - 4000(t+8)^-2. At what time are weekly sales falling the fastest?

Question

anonymous · Answer

I already found the derivative of the equation to be f'(t) = -(1000 t)/(t+8)^3

anonymous · Answer

@zepdrix @Luigi0210

zepdrix · Answer

Mmmm if I'm interpreting this correctly it seems....
Ok so they want to know when the slope is the most negative of this function.
The derivative gives us slope at certain points,
and we want to `minimize` slope, so we'll want to find critical points not of our function, but of our derivative function.

So we want to take a second derivative, and set it equal to zero and find out where the slope is the lowest and highest and such :d

zepdrix · Answer

$$\Large\rm f(t)=1000(t+8)^{-1}-4000(t+8)^{-2}$$$$\Large\rm f'(t)=-1000(t+8)^{-2}+8000(t+8)^{-3}$$$$\Large\rm f''(t)=2000(t+8)^{-3}-24000(t+8)^{-4}$$

zepdrix · Answer

Getting a common denominator,
$$\Large\rm 0=\frac{2000(t+8)-24000}{(t+8)^4}$$