The daily cost of producing x high performance wheels for racing is given by the following​ function, where no more than 100 wheels can be produced each day. What production level will give the lowest average cost per​ wheel? What is the minimum average​ cost?C(x)=0.09x^3 - 4.5x^2 + 180x; (0,100]

Answer:The production of wheel per day is 74 which gives lowest average cost per wheel.The minimum average cost is $168.72.Step-by-step explanation:Given function of average cost is[tex]C(x)= 0.03x^3-4.5x^2+171x[/tex]Differentiating with respect to xC'(x)= 0.09 x² -9.0 x+171Again differentiating with respect to xC''(x) = 0.18 x -9.0To find the minimum average cost, first we have to set C'=0.The function's slope is zero at x=a, and the second derivative at x=a isless than 0, it is critical maximum.greater than 0,  it is critical minimum.Now ,C'=0⇒ 0.09 x² -9.0 x+171=0⇒x = 74.49, 25.50[tex]C''(x)|_{x=74.49} = 0.18 (74.49)-9.0=4.41>0[/tex][tex]C''(x)|_{x=25.50} = 0.18 (25.50)-9.0=-4.41<0[/tex]Therefore at x= 74.49≈ 74, the average cost is minimum.The production of wheel per day is 74 which gives lowest average cost per wheel.The minimum average cost [tex]C(x)= 0.03x^3-4.5x^2+171x[/tex]                                                       [tex]=(0.03 \times 74^3)-(4.5 \times 74^2)+(171\times 74)[/tex]                                                       =168.72[Assume the average cost is in dollar]The minimum average cost is $168.72.