Hi there. I'm working through "Forgotten Calculus" (Barbara Lee Bleau) and am stuck on an optimisation problem. I'll give the question, show my attempt, and then reveal her (stated, not worked) answer. Would appreciate guidance.

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a) the question

The price equation for a new product is

p= 1,600 - 4x

where p = price and x = # of units produced.

The cost function is 40x + 2,000.

Find the maximum profit and the price which needs to be charged to achieve it.

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b) I'm using an approach which has worked well with a number of similar-looking problems.

What's to be maximized? Profit

Profit = revenues - costs

= (units * price) - (units * costs)

= (x (1600-4x)) - (x (40x + 2000))

= (1600x - 4x^2) - (40x^2 + 2000x)

= 1600x - 4x^2 - 40x^2 - 2000x

= -400x - 44x^2

Even though I'm already struggling to see how this product will make anything but losses, I seek out maxima via the first derivative

Profit ' = -400 -88x

0 = -400 - 88x

400 = -88x

-400 = 88x

-4.545454 = x

Plugged back into the original function, it means the more units produced, the more money lost. Making a profit seems impossible as it would mean selling a negative number of units.

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Her answer: "A price of $824 leads to a maximum profit of $148,544". I cannot reach this.

Where am I going wrong?

Ken