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Show that the objective function $P=A x+B y+C$ is either increasing, decreasing or remains the same when it is evaluated along the line $y=m x+b .$ (Hint: Substitute for $y$ in the objective function and apply the first derivative test.)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

Baylor University

University of Nottingham

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. Wish the objective function P equals X plus B. Y, pussy. Mhm. It's gonna have to either be increasing or decreasing or remains the same when it is evaluated along the line. Why it goes max must be. So we're going to substitute Why? And for the objective function and apply the first derivative test. So this is gonna be a maximus B. Yeah. And we see that this is going to become B. Mx plus B. B. Um We take the first derivative to do the first derivative test. P prime community getting A plus B. Um top M. And then these are constant. So they go to zero. We set this equal to zero. Were able to find um the fact that they're going to be increasing, decreasing or remaining the same when evaluated along this line. So it's gonna be our final answer.

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