help, anyone?

[TexaSASS]

P(x) = -x^3 + 12x^2 - 21x + 100, x ≥ 4 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

I have this problem and I am stuck. I don't know what to do. My sister did it and she got 7 hundred thousand, but I don't know if that is right. I tried to work backwards and it is just not working for me, can anyone offer any help?

Thanks!

 

[skeeter]

your sister is correct.

P(x) = -x^3 + 12x^2 - 21x + 100

P'(x) = -3x^2 + 24x - 21

set P'(x) = 0 ...

-3x^2 + 24x - 21 = 0
-3(x^2 - 8x + 7) = 0
-3(x - 1)(x - 7) = 0

since x > 4, x = 7 is a possibility ...

P"(x) = -6x + 24
P"(7) < 0, so x = 7 yields a maximum.