The total monthly cost (in dollars) incurred by a certain guitar manufacturer in manufacturing [Math Processing Error]x units of its Professional Series guitars is given by the function [Math Processing Error]C(x)=0

We will divide the cost function [Math Processing Error]C(x) by [Math Processing Error]x to get the average cost function:

[Math Processing Error]C¯(x)=C(x)x=0.001x2+150x+20250x=0.001x+150+20250x

a. Thus, the average cost function is [Math Processing Error]C¯(x)=0.001x+150+20250x.

We will set the derivative of [Math Processing Error]C¯(x) equal to [Math Processing Error]0 to get the smallest average production cost.

Differentiating [Math Processing Error]C¯(x):

[Math Processing Error]C¯(x)=0.001x+150+20250xC¯′(x)=0.001−20250x2

Equating [Math Processing Error]C¯′(x) to [Math Processing Error]0 and solving for [Math Processing Error]x:

[Math Processing Error]C¯′(x)=00.001−20250x2=0x2=202500.001x=20250000x=4500

b) Therefore, [Math Processing Error]4500 units is the production level that will result in the smallest average production cost.