See of x equals 78 thousand four hundred plus eight hundred x plus x squared where C of X is the total cost and X is e production level a we want to find the production level that will minimize the average cost and also B find the minimum average cost we need to be careful here because we're asked to minimize the average cost not the total cost so for the first step we want to find the average cost function where the average cost function is equal to the total cost function divided by X where C of X represents the total cost and next represents the production level so the average cost function is equal to C of X divided by X or in our case 78,000 400 800 X plus x squared divided by X so because we're dividing by a monomial we want to divide each term in the numerator by X so I have seventy thousand four hundred divided by X plus 800 X divided by X plus x squared divided by X so the average cost function is equal to seventy eight thousand four hundred divided by X plus this would just be eight hundred plus this would just be X and now to minimize the average cost function our first step is to find the critical numbers for this function which occurs where its derivative is equal to zero or undefined or when the average cost and the marginal cost are equal we're going to use method a and determine when the derivative of this function is equal to zero or undefined notice here we have X to the first in the denominator so if we move this up the numerator it would change the sign of the exponent so we can...