Question

Can some one help with this one? I am new to derivatives.

A company finds that the profit 𝑃(𝑥) where 𝑥 represents thousands of units, is given by
𝑃(𝑥)=`−x^3+9x^2−15x−9`

If the company can only make a maximum of 6000 units, what is the absolute maximum profit?

Answer 1

At 5,000 units, the absolute maximum profit is `P = 16`.

Explanation:

`P(x) = −x^3+9x^2−15x−9`

`x in [0,6]`

1st derivative to optimise :

`P'(x) = −3x^2+ 18x −15 = -3 (x - 5) (x - 1) = 0`

`P' = 0 implies x = 1,5`

2nd derivative test for max/min :

`P''(x) = −6x+ 18`

• `{(P''(1) = 12 > 0),(P''(5) = - 12 < 0):}`

So profit is maximised at `x = 5, P = 16`

That is, at 5,000 units, the profit is `16`.