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In calculus, extrema refer to the highest or lowest values (peaks or troughs) that a function attains. These are commonly referred to as maximum and minimum points. To spot an extremum, one can use the first and second derivative tests, or investigate obvious candidates, such as in the case of the function
\(f(x)=4-|x|\).

For example, as seen in the exercise, the extremum of our function occurs at \(x=0\), the point where the function reaches its maximum value of 4. Determining the extremum is essential, as these points often represent critical behavior of the function in real-world applications, such as finding the optimal solution to a problem.

In finding derivatives, extrema can give us important information. At these points, the derivative typically equals zero, indicating a horizontal tangent line. However, in the uniqueness of absolute value functions, as with \(f(x)=4-|x|\), the extremum may exist at a point of non-differentiability.

Piecewise functions are defined by different formulas in different parts of their domains. In the context of the textbook problem, the absolute value function is a classic example of a piecewise function.

The function \(f(x)=4-|x|\) is broken down into \(f(x)=4-x\) for \(x>0\) and \(f(x)=4+x\) for \(x<0\). This two-part definition allows us to address the function in each domain separately when computing the derivative. In practice, piecewise definitions are like instructions that tell us which formula to use based on the input value.

When working with piecewise functions, special attention must be given to the points where the definition switches, as these can often be points of nondifferentiability or other interesting behaviors. They are pivotal for understanding the overall shape and properties of a function.

Non-differentiability occurs in a function when the derivative does not exist at a certain point. For absolute value functions, this commonly happens at 'sharp points' or 'corners.' These are points where the function abruptly changes direction, like at \(x=0\) for \(f(x)=4-|x|\).

At these sharp points, the slope of the tangent cannot be defined because the function approaches the point with different slopes from the left and the right side. The derivative, which represents the slope of the tangent line to the curve at a point, cannot jump from one value to another; it must be consistent in the immediate vicinity of the point in question.

Therefore, sharp points are inherently problematic for finding derivatives. Students must recognize that at such points, the function can have a maximum or minimum value and yet fail to have a derivative, which contrasts with other smooth points of extrema where the derivative is typically zero.