Find all extreme values (if any) of the given function on the given interval. Determine at which

step 1 So, this is given fx, so f -xx equal to x minus 1. So, for finding critical number put f -dash -

Find all extreme values (if any) of the given function on...

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then the function must attain both a maximum and a minimum value on that interval. This is fundamental in calculus because it guarantees that once the function's behavior is understood over a closed interval, one can be certain that there are global extrema (both highest and lowest function values) within that region.

Critical Points and Derivatives

A critical point of a function is a point where the function's derivative is zero or undefined. In calculus, critical points are important because they are potential locations for local maxima or minima. By computing the derivative and setting it equal to zero, one can locate these candidate points, which then need further analysis to determine if they correspond to extreme values.

Closed Interval Method

The Closed Interval Method is a systematic approach to finding the global maximum and minimum values of a continuous function over a closed interval. It involves calculating the function’s value at all critical points within the interval as well as at the endpoints of the interval, and then comparing these values to determine the extrema.

Evaluation of Function Values

After identifying all critical points and considering the endpoints, the next step is to evaluate the function at these points. The function values obtained are then compared to find the least (minimum) and greatest (maximum) values. This methodical evaluation is essential to accurately determine the extreme values of the function over the specified interval.

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