Question

A company estimates that it will sell ????(x) units of a product after spending x thousand dollars on advertising, as given by ????(x)=−5x3+235x2−3400x+20000,10≤x≤40. (A) Use interval notation to indicate when the rate of change of sales ????′(x) is increasing. Note: When using interval notation in WeBWorK, remember that: You use 'I' for [infinity] and '-I' for −[infinity], and 'U' for the union symbol. If you have extra boxes, fill each in with an 'x'. ????′(x) increasing:

Answer 1

Answer:

$(11\frac{1}{3},20)$

Step-by-step explanation:

If the Number of Sales of x units, N(x) is defined by the function:

$N(x)=-5x^3+235x^2-3400x+20000,10\leq x\leq 40$

$N'(x)=-15x^2+470x-3400\\When -15x^2+470x-3400=0\\x=20, x=11\frac{1}{3}$

Next, we text the critical points and the end points of the interval to see where the derivative is increasing.

$N'(10)=-200\\N'(11\frac{1}{3})=0\\N'(19)=115\\N'(20)=0\\N'(40)=-8600$

Thus, the rate of change of sales $N'(x)$ is increasing in the interval $(11\frac{1}{3},20)$ on 10≤x≤40.