Several Part Calc Question

Question

The total revenue and total cost functions for the production and sale of x TV's are given asR(x)=140x−0.4x2andC(x)=3800+21x.

(A) Find the value of x where the graph of R(x) has a horizontal tangent line. x values is 

(B) Find the profit function in terms of x.

P(x) = 

(C) Find the value of x where the graph of P(x) has a horizontal tangent line.

x values = 

(D) List all the x values of the break-even point(s).

If there are no break-even points, enter 'NONE'.

List of x values = 

Answer 1

R(x) = 140x -0.4x^2

R'(x) = 140-.8x=0

x = 140/.8 = 1750

x=175 is where the tangent line to the Revenue function is horizontal

175 is the revenue maximizing output level

C(x) = 3600 +21x

P(x)= R(x)-C(x) = 140x-0.4x^2-(3600+21x) = -0.4x^2 +119x -3600

P'(x) = -0.8x +119=0

x = 119/.8 = 148.75 when the Profit function has a tangent line with zero slope

148.75 is the profit maximizing output level

breakeven points are when P(x)=0

0.4x^2 -119+3600=0

x =119/.8 +/- (1/.8)sqr(119^2 -4(.4)(3600))

x = about 34 or about 263 TV sets for zero profit, zero loss