In Problems 37-64, find the absolute maximum value and...

In Problems $37-64$, find the absolute maximum value and absolute minimum value of each function on the indicated interval. Notice that the functions in Problems $37-58$ are the same as those in Problems $13-34$ $$ f(x)=\left\{\begin{array}{ll} x^{2} & \text { if }-2 \leq x<1 \\ x^{3} & \text { if } \quad 1 \leq x \leq 2 \end{array}\right. $$

In Problems $37-64$, find the absolute maximum value and absolute minimum value of each function on the indicated interval. Notice that the functions in Problems $37-58$ are the same as those in Problems $13-34$
$$ f(x)=\left\{\begin{array}{ll} x^{2} & \text { if }-2 \leq x<1 \\ x^{3} & \text { if } \quad 1 \leq x \leq 2 \end{array}\right. $$

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest (maximum) and lowest (minimum) values that a function reaches over a specified interval. These values are found by evaluating the function at all critical points and at the endpoints of the interval, ensuring that every potential candidate for an extreme value is considered.

Piecewise Functions

A piecewise function is one that is defined by different expressions over different portions of its domain. When working with such functions, it is crucial to analyze each sub-function separately, particularly at the points where the definition changes, to ensure that all possible extreme values are correctly identified.

Critical Points

Critical points are points within the domain of a function where the derivative is zero or does not exist. They are significant in optimization problems because an absolute maximum or minimum can occur at these points. For piecewise functions, special attention is required at the boundaries between pieces, where the derivative might not be defined.

Endpoint Analysis

When finding absolute extrema on a closed interval, endpoints must be evaluated along with the critical points. According to the Extreme Value Theorem, a continuous function on a closed interval is guaranteed to have both an absolute maximum and an absolute minimum, making the endpoints potential candidates for these extreme values.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, it will have both an absolute maximum and an absolute minimum on that interval. This theorem is fundamental when analyzing kernels of continuity and guaranteeing that a solution exists within the given interval.

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