Question

A company estimates that it will sell N(x) hair dryers after spending $x thousand on advertising as given by: N(x)=-0.25x^4+13x^3-180x^2+10,000 for 15<=x<=24 When is the rate of change of sales increasing, decreasing?(Hint: Find N'(x) which is the rate of change, then find N"(x) to check when N'(x) increasing or decreasing)

Answer 1

If

N(x) = -0.25x^4+13x^3-180x^2+10000

then

N ' (x) = ???

Answer 2

-1x^3+39x^2-180

Answer 3

good

Answer 4

now what is N '' (x)

Answer 5

-3x^2+78x

Answer 6

very nice

Answer 7

Im almost a pro haha

Answer 8

you're definitely a pro

Answer 9

Now we need to find the roots of N '' (x) = -3x^2+78x

N '' (x) = -3x^2+78x

0 = -3x^2+78x

-3x^2+78x = 0

-3(x-26) = 0

x - 26 = 0

x = 26

So x = 26 is a possible point of inflection

Answer 10

If x = 25 (a value to the left of x = 26), then

N '' (x) = -3x^2+78x

N '' (25) = -3(25)^2+78(25)

N '' (25) = 75

This tells us that the rate of change is increasing on the interval (-infinity, 26)

Answer 11

If x = 27, then

N '' (x) = -3x^2+78x

N '' (27) = -3(27)^2+78(27)

N '' (27) = -81

which means that the rate of change is now decreasing on the interval (26, infinity)

Answer 12

if we restrict the interval to 15 <= x <= 24 (just saw this part), then the rate of change of the sales is increasing because 15 <= x <= 24 is definitely in the interval (-infinity, 26)

Answer 13

okay, so the increasing interval is set

Answer 14

since the derivative is increasing throughout the interval 15 <= x <= 24, there are no intervals where it is decreasing

Answer 15

so I would say NA or none or something like that

Answer 16

oh wow that was a lot easier than I thought!

Answer 17

yeah not too bad

Answer 18

this is only true because 15 <= x <= 24 is all below 26 where everything is increasing

if you stretched it from 15 <= x <= 24 to 15 <= x <= 34, then it would be a different story

Answer 19

I want to know why n'(x) isn't -1x^3+39x^2-360x?