A cost function of the form C(x) = 1/2x² reflects diminis...

Question

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

Accepted Answer

A company's production costs are modeled by the function C of X equals 3x2 + 2X, where C of X is the total cost in dollars and X is the number of units produced. Determine and graph the marginal cost function. Now, this graph on the right is the marginal cost function, but we'll get to how we can find this in a moment. Let's first find the marginal cost function itself. Now, the marginal cost function is given by the derivative of our cost function C of X. So we are looking for C of X. Now we know we have 3 X2 + 2 X. To solve this derivative, we can make use of the power rule, which tells us if we have The derivative of something X to the and power. Its respected derivative is NX raises the N minus 1 power. So, we have 3 X2 + 2X. This turns out to be 3, multiplied. By 2 X Plus 2. Which gives us 6 X plus 2. So this is the function we want to graph. And we noticed this is a linear function. Now, because this is linear, We can use our slope intercept form. Where our slope M is equivalent to 6. And our Y intercept B is equivalent to 2. We can now find 2 example points on her graph. If we say X equals 0, And X equals 1, plugging in these values gives us Y equals 2. And Y equals 8. Give us the points, 02 and 18. Now, the graph on the right here has a linear function. With points of 02. And 1/8, and a black line between the two points. So this graph, along with our function 6 X + 2 are the answers to our problem. OK, I hope to help you solve the problem. Thanks for watching. Goodbye.

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

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