Find the maximum profit and the number of units that must be...

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, $R(x),$ and $\cos t, C(x)$ are in dollars.
$$R(x)=50 x-0.5 x^{2}, \quad C(x)=10 x+3$$

Key Concepts

Profit Function

The profit function represents the difference between the revenue function and the cost function. It is a fundamental concept in economics and business math, where determining profit requires subtracting total costs from total revenues. This function is used to analyze how changes in units produced and sold affect overall profitability.

Quadratic Functions

Quadratic functions are polynomial functions of degree two which often model real-world scenarios in economics, such as revenue, cost, and profit. Their graphs form parabolas, and the vertex of a parabola represents the maximum or minimum value of the function, an important concept when seeking to optimize profit.

Optimization via Calculus

Optimization using calculus involves finding the maximum or minimum values of a function by calculating its derivative, setting it equal to zero to find critical points, and using the second derivative to confirm the nature of these points. This process is essential in determining the quantity that maximizes profit in business scenarios.

Transcript

00:02 Okay, on this one, number 24, we have a total cost, a total revenue, and then we have a profit definition from the book.

00:11 That's revenue minus cost, the money coming in, minus the money coming out.

00:16 Okay, so we're going to use these to build a profit equation.

00:21 That profit equation is going to be something that we can use then to maximize it.

00:25 So let's start off by getting our profit, p of x, i'll figure out.

00:28 So p of x is going to be equal to r of x, which is...

00:32 50 x minus 0 .5x squared minus this whole deal.

00:39 Let's not mess up our negative signs.

00:45 Okay.

00:47 I'm going to work through it.

00:48 This over here is going to become minus 10x, minus 3.

00:51 That's the easy part to mess up.

00:52 These will remain the same.

00:55 So let's go ahead and combine our like terms.

00:58 We're going to have 40x.

01:01 This stays the same.

01:05 And then we're just left with the minus 3.

01:07 The end.

01:09 All right.

01:10 So why do we take the derivative of this? well, we're trying to figure out when this equation, which probably looks something like a parabola, there's a magic point, right? it's the highest point.

01:20 And at that point, there's a corresponding x value.

01:23 Well, this also happens to be the one spot where the slope is equal to zero.

01:28 And the slope, as we found out in this type of equation, is equal to the derivative.

01:33 Okay, that's p of x.

01:36 So when that's zero, meaning a horizontal line, then we can go ahead and assume that that's going to be the maximum...

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