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Suppose $f$ is continuous for $a \leq x \leq b$. Show that the function attains both a maximum and minimum value. That is, there are at least two numbers, $c_{1}$ and $c_{2},$ between $a$ and $b,$ such that, $f\left(c_{1}\right)=M$ and $f\left(c_{2}\right)=N,$ where $f(x) \leq M$ and $f(x) \geq N,$ for all $x$ such that $a \leq x \leq b .$ This is called the Extreme Value Theorem

02:24

Fahad P.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Missouri State University

Campbell University

Harvey Mudd 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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Functions Let the function…

let F be a continuous function defined on a closed interval A B parquet. We talk about what theorem guarantees the existence of an absolute maximum value and and an absolute minimum value for F. In part B. We talk about what steps are necessary to find those extreme values of the function. So in part A we are we are talking about the extreme value theory which we have here and that theory mistakes that if a function F is continuous on a closed interval baby, then F attains an absolute maximum value F. Of C. And an absolute minimum value F. F. D. At some point. And some numbers see in the in the interval A. B. So this is your guarantee guaranteeing us the function attains it's extreme valleys on a close interval and the function gotta be continued to that purple. We talk about the steps needed to find those C and D. For which the images are correspondingly the absolute maximum value and the absolute minimum body of the function. That's what we call the close interval method. And under the assumptions of the extreme value theory and there is a function is continuous and close interval. What we do first step one is find the values of F at the critical numbers of F in the interval A be open interval that is We find the first derivative of F and all the values in dangerous. A. B open interest that is excluding the the end points for which derivative doesn't exist or Is equal to zero. All those minds got to be considered. Step wonder is we get to the way the function at those critical numbers that are within the interval. Ape The step two is find the values of the function at the end points of the interval that is calculates f f a n f f B. The images of the importance and step three is find or to the largest value of the value calculated in steps one of them to the largest value what or better. The larger of the values noticed of the values from steps one and two. The largest of those values is the absolute maximum value of F. Only closing to vote baby and the smallest of tom's valleys is the absolute minimum value function on your clothes in general. So will you call we make a summary under the hypothesis of the extreme value theory. There is a function is continuous on a closed interval A B Step one, we find the values of the function at the critical numbers of f years of F. That's very currently of death. We find the critical numbers of F in the Interpol the way that is the values in that interval for which the relative doesn't exist or is equal to zero. So we got to calculate the first derivative and solve the equation after every difficulty and inspect whether the derivative is not defined. And so at some points inside the that all those points that that is all the critical numbers of Fn Maybe we got to have a function there In step two. Either way to function at the end points of the Internet, that is. We find the values F. Of A and F. B. And then we issued the largest of the values calculated in steps one and two. And that the absolute maximum value of the function over interval maybe which is the smallest of the values calculated in steps one and two. And that's the absolute minimum value, the function over the close interval. So these are the steps we got to follow each time we want to find the extreme violence of the continuous function on the claws into. Mhm.

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