Can anyone tell me how to do the maximize part?

Question

The Pear company sells pPhones. The cost to manufacture x pPhones is C(x)=-19x^2+48000x+19934 dollars (this includes overhead costs and production costs for each pPhone). If the company sells x pPhones for the maximum price they can fetch, the revenue function will be R(x)=-22x^2+120000x dollars.

How many pPhones should the Pear company produce and sell to maximimze profit? (Remember that profit=revenue-cost.)

Answer 1

P(x)=R(x)-(C)= -22x^2+120,000x -(-19x^2+48,000x+19934)

= -3x^2- 72,000x -19934

P'= -6x -72000=0

x=-12,000

or another method without calculus

-3x^2 -72,000x-19934

complete the square add half 24000 squared

=-3(x^2+24000x +12000^2) -19934 +12000^2

=-3(x+12,000) ^2 +143,981,166 ln vertex form: a(x-h)+k with (h,k)=vertex=(-12000,about 144 million)

but a negative production level usually doesn't make sense

there may be a mistake in the problem

or it may be a trick question with x=0 as the answer if domain is [0, infinity)