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#.

#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#
So profit is maximised at #x = 5, P = 16#
That is, at 5,000 units, the profit is #16#.
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Answer 2

Of course! Please go ahead and ask your question about derivatives, and I'll do my best to help you understand.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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