Find the minimum and maximum values of the function on the given interval by comparing values at

step 1 We are going to locate our absolute extremist. So let's go ahead and find the critical numbers.

Find the minimum and maximum values of the function on the...

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Key Concepts

Extreme Value Theorem

This theorem guarantees that a continuous function on a closed interval attains both absolute minimum and maximum values. It forms the basis for evaluating the function at critical points and endpoints to determine these extreme values.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or undefined. These points are important because they are potential locations where the function's rate of change shifts, possibly creating local minima or maxima, which in turn can be candidates for absolute extrema on a given interval.

Derivative Calculation

Computing the derivative of the function is a fundamental step in optimization. The derivative provides the rate of change of the function, and setting the derivative equal to zero is the standard procedure to locate critical points. This process helps in identifying where the function may attain a local maximum or minimum.

Endpoint Evaluation

For functions defined on a closed interval, the endpoints must be evaluated as potential candidates for global extrema. Even if critical points exist in the interior of the interval, absolute extreme values might occur at the boundaries, so comparing function values at both the critical points and endpoints is essential.

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