In Exercises 17-36, locate the absolute extrema of the function on the closed interval. y=3-|t-3

step 1 In this question we call that the absolute extremer occurs only at either the critical number or

In Exercises 17-36, locate the absolute extrema of the...

Key Concepts

Absolute Value Functions

Functions that include absolute value expressions often require special treatment because they can be defined piecewise. Understanding the properties of absolute value functions is important in finding extrema since it may involve analyzing different cases based on the sign of the expression inside the absolute value, ensuring that all possible variations of the function over the domain are considered.

Critical Points

Critical points are points in the domain of a function where the derivative is zero or does not exist. Since these points represent potential locations of local maxima or minima, they are key candidates when determining the absolute extrema of a function, especially when combined with the evaluations at the interval endpoints.

Extreme Value Theorem

The Extreme Value Theorem states that any continuous function defined on a closed interval will have both an absolute maximum and an absolute minimum. This theorem is fundamental in calculus when searching for extrema because it guarantees the existence of both types of extreme values over a closed bounded interval.

Closed Interval Method

This method involves evaluating the function at all critical points within the interval as well as at the endpoints of the interval. By comparing these values, one can determine the function’s absolute maximum and minimum. The process is essential in optimization problems where the domain is restricted to a finite interval.

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