A one-product firm estimates that its daily total cost...

A one-product firm estimates that its daily total cost function (in suitable units) is $C(x)=x^{3}-6 x^{2}+13 x+15$ and its total revenue function is $R(x)=28 x .$ Find the value of $x$ that maximizes the daily profit.

Key Concepts

Profit Function

The profit function is a key concept in economics that represents the difference between total revenue and total cost. It is used to determine the profit earned by a firm at various levels of production, and formulating this function is essential for any analysis of profit maximization.

Profit Maximization

Profit maximization is the objective of determining the level of output that yields the highest possible profit for a firm. This process involves analyzing how changes in output affect profit and identifying the point at which profit is maximized.

Calculus-Based Optimization

Calculus-based optimization involves finding the maximum or minimum values of functions using derivatives. The technique is widely applied across economics and business to determine optimal production levels and other decision variables.

First Derivative (Critical Points)

Taking the first derivative of the profit function with respect to output helps find critical points where the rate of change of profit is zero. These critical points are candidates for local maxima or minima in the profit function.

Second Derivative Test

Once the critical points are found, the second derivative test is used to determine the concavity of the profit function at these points. A negative second derivative indicates that the function is concave down at that point, confirming a local maximum.

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