A profit -maximizing firm has the total-cost function c= x^2 + 2x and sells into a competitive market on which the price is $10. what output should it produce?hint: Find the derivative and check for local maxima and minima

Answer:The output produce is 4.Step-by-step explanation:Given : A profit -maximizing firm has the total-cost function [tex]c= x^2+2x[/tex] and sells into a competitive market on which the price is $10.To find : What output should it produce?  Solution : The total-cost function [tex]C(x)= x^2+2x[/tex] The revenue function is price into number of item,So, The revenue function is [tex]R(x)=10x[/tex]The profit function is given by,[tex]P(x)=R(x)-C(x)[/tex][tex]P(x)=10x-(x^2+2x)[/tex][tex]P(x)=10x-x^2-2x[/tex][tex]P(x)=8x-x^2[/tex]The derivative of the profit function,[tex]P'(x)=8-2x[/tex] Equate it to zero to get output,[tex]8-2x=0[/tex] [tex]2x=8[/tex]   [tex]x=4[/tex] For maxima/minima we find the second derivative,[tex]P''(x)=-2[/tex] As [tex]c''(x)<0[/tex] it is a local maxima. Therefore, The output produce is 4.