Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that​ revenue, Upper R left parenthesis x right parenthesisR(x)​, and​ cost, Upper C left parenthesis x right parenthesisC(x)​, of producing x units are in dollars. Upper R left parenthesis x right parenthesisR(x)equals=4 x4x​, Upper C left parenthesis x right parenthesisC(x)equals=0.01 x squared plus 0.3 x plus 40.01x2+0.3x+4

Accepted Solution

Answer:185 unitsStep-by-step explanation:Given,The revenue function is,[tex]R(x) = 4x[/tex]Cost function,[tex]C(x) = 0.01x^2 + 0.3x + 4[/tex],Where,x = number of units produced.Thus, profit = revenue - cost[tex]P(x) = 4x - ( 0.01x^2 + 0.3x + 4) = -0.01x^2 + 3.7x - 4[/tex]Differentiating with respect to x,[tex]P'(x) = -0.02x + 3.7[/tex]Again differentiating with respect to x,[tex]P''(x) = -0.02[/tex]For maxima or minima,P'(x) = 0,[tex]-0.02x + 3.7x =0[/tex][tex]-0.02x = -3.7[/tex][tex]\implies x = \frac{3.7}{0.02}=185[/tex]For x = 185,P''(x) = negative,Hence, for maximising the profit 185 units must be produced.