Consider the following. C = x^3 - 6x^2 + 13x Use the cost...

Consider the following. C = x^3 - 6x^2 + 13x Use the cost function to find the production level at which the average cost is a minimum. x = For this production level, show that the marginal cost and average cost are equal. Use a graphing utility to graph the average cost function and verify your results. marginal cost $ average cost $

Transcript

00:01 Hello everyone, in this problem we are given with the cost function c of x as x cube minus 6 x square plus 13 x.

00:14 So, now we need to use this cost function to find the production level at which the average cost is minimum.

00:23 So, in order to calculate the average cost of this function, average cost will be the cost function by x, that is x cube minus 6 x square plus 13 x divided by x.

00:55 So, on dividing this we have the value to be average cost of x to be equal to x square minus 6 x plus 13.

01:09 Now, differentiating this with respect to x, so we get d of ac of x to be divided by dx which will be equal to, so now differentiating this it will be 2x minus 6.

01:24 So, now taking this value to 0 in order to find the minimum value, so it will be 2x minus 6 which is equal to 0.

01:39 So, from this we can write 2x as equal to 6.

01:43 So, from this we get the value of x to be 6 by 2 which is 3.

01:48 So, now again on differentiating the average cost with respect to x, so we get d square ac of x to be divided by dx square will be equal to, again differentiating with respect to x it will be 2.

02:17 So, now this value is greater than 0 and at the point x equal to 3 this value will also be greater than 0.

02:30 So, therefore we can say that the average cost, sorry, average cost is minimum when x equal to 3.

02:54 So, therefore now we can calculate its marginal cost.

03:00 So, marginal cost, let us take this to be mc of x and this will be obtained by differentiating the cost function with respect to x that is d by dx of c of x...

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