Video instructions and help with filling out and completing How Form 5495 Limitation

Instructions and Help about How Form 5495 Limitation

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 also write the average cost function as seventy eight thousand 401 plus eight hundred plus X and now we'll find the derivative function on the next slide so the derivative of the average cost function would be equal to the derivative of the first term we're going to multiply by negative 1 so we have negative 70 8400 and then subtract 1 from the exponent that would be X to the power of negative 2 the derivative of 800 would be 0 the derivative of X would be 1 so we have plus 1 so if we want to we could write this as negative 78,000 400 divided by x squared if we move this back to the denominator it'll change the sign of the exponent then we have plus 1 notice how this would be undefined when x equals 0 but when x equals 0 the production level of zero and the average cost function is undefined so we'll set this equal to zero and solve for x to determine the critical numbers let's go ahead and add this fraction to both sides of the equation that would give us 1 equals 78 thousand four hundred divided by x squared and now here we can multiply both sides of the equation by x squared or make this a proportion by putting this over one and cross multiplying notice 1 times x squared must equal 1 times 78,000 400 so again we'd have x squared equals 70 thousand four hundred and now we'll square root both sides of the equation but again since X represents the production level we know X can't be negative so we'll in concerned about the principal square root or the positive square root so now we'll go to the calculator and we have the square root of seventy eight thousand four hundred so x equals 280 so our only critical number is x equals 280 so there should be the production level that will minimize the average cost but to be sure we should use either the first or second derivative tests to make sure the function is minimized rather than maximized at this x value let's go and use the second derivative test to determine if the function is concave up or down at x equals 280 so here's the first derivative so the second derivative would be equal to so to find the derivative of the first term we're going to multiply by negative 2 that's going to give us 150 6800 X to the power of negative 3 the derivative of 1 would be 0 let's go ahead and write this as 150 6800 divided by X cubed so notice how when X is equal to 280 this would be positive or greater than zero which means the function will be concave up which is good news because that means at this critical number we have a low point or a minimum function value so to answer Part A a production level of 280 will minimize the average cost and now for Part B to find the minimum average cost will evaluate the average cost function here when x equals 280 let's go ahead write the average cost function back in the form of 78,000 400 divided by X plus 800 plus X so the average cost when x equals 280 would be 78 thousand four hundred divided by 280 plus 800 plus 280 now we'll go back to the calculator notice how this sum here would be 1080 so the minimum average cost is one thousand three hundred sixty dollars I hope you found this helpful