DoThatMath all math,Calculus Understanding The Extreme Value Theorem, Critical Numbers, Absolute Minimum & Maximum

Understanding The Extreme Value Theorem, Critical Numbers, Absolute Minimum & Maximum

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In this  post, we will be discussing the Extreme Value Theorem, Critical Numbers, Absolute Minimum & Maximum. This theorem is important in Calculus and Analysis. It basically states that given a function defined on a closed interval, there must exist a maximum and minimum value of the function on that interval. We will be discussing what these terms mean and how to find them.


The Extreme Value Theorem

The Extreme Value Theorem tells us that given a function defined on a closed interval, there must exist a maximum and minimum value of the function on that interval. In other words, if you have a function f(x) defined on the closed interval [a,b], then there must be some point c in the interval where f(c) is either the maximum or minimum value of the function on that interval.

The theorem does not tell us whether the maximum or minimum value occurs at an interior point or endpoint of the interval. It also does not tell us how many critical points (points where the derivative is 0 or undefined) there are on the interval. We will discuss how to find these things in later blog posts.

Critical Numbers

A critical number of a function is a number at which the derivative is 0 or undefined. In other words, it is a point where the slope of the tangent line is 0 or undefined. Finding critical numbers can help us find the absolute minimum and maximum values of a function.

Absolute Minimum
The absolute minimum of a function is the lowest possible output value of the function. To find the absolute minimum of a function, you need to find all local minima (critical points where the derivative changes from positive to negative) and then compare these local minima to find which one corresponds to the absolute minimum.

Absolute Maximum
The absolute maximum of a function is the highest possible output value of the function. To find the absolute maximum of a function, you need to find all local maxima (critical points where the derivative changes from negative to positive) and then compare these local maxima to find which one corresponds to the absolute maximum.

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