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Use the Mean Value Theorem to prove the First Derivative Test. Hint: let $x_{1}<x_{2}$ be any two points in the interval in question. Apply the Mean Value Theorem using these points to deduce that $f\left(x_{2}\right)-f\left(x_{1}\right)=\left(x_{2}-x_{1}\right) f^{\prime}(c)$.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

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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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Use the Mean Value Theorem…

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Verify that the function s…

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Verify that $f$ satisfies …

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Mean Value Theorem Conside…

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here is probably going to use the mean value theorem in order to prove the first derivative test. So we're gonna let X one less the next to be any two points in the interval. We want to apply the mean value theorem using these points to deduce that ffX two minus F of X one Is able to X 2- Excellent Times F. Primacy. Um So what we see is that if we go back to this, this that we're going to call X two and this is going to be X one Makes this X two. That's excellent. With that in mind. We can multiply Both the top and the bottom by X 2 -11. Bring it over here. Yeah, yeah. When we do that every month right here and we end up getting this result which we wish to show.

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