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Find two numbers whose sum is 100 such that their product is as large as possible. Justify your conclusion!

$$(50,50)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Harvey Mudd College

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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for the following problem, we want to find two numbers whose sum is 100 such that their product is as large as possible. So we have one number X plus another number, Y is equal to 100. And then we want to maximize xy the way that we do this is by recognizing the X actually will do that. Why Take up to 100 -X. The reason why that's important to know is because now we can replace why With 100- X. So that's why now becomes 100 100 minus X. So we have 100 X minus X squared. Another reason we do that is because now we have an actual function that can be maximized And we see this occurs when X is equal to 50. So in fact says he got a 50, That means why is also equal to 50. So we end up getting 50 50 is our final answer. And we can verify this because we end up getting an area Of 2500, which is greater than any other area we can get, as evidenced right here.

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