91Ó°ÊÓ

When studying the behavior of functions, \textbf{critical points} are crucial in understanding where the function may change in direction, has horizontal tangents, or is not differentiable. These points are locations on the graph of a function at which the derivative equals zero or is undefined. To find critical points, you first need to take the derivative of the function.

In the context of the given function, the fact that it includes an absolute value makes it piecewise, and which leads us to the concept that at the point where the function changes its formula - here at \( t = 3 \) - a critical point is identified because the derivative of the absolute value function is not defined at this point. This is where the behavior of the function changes, indicating potential minima or maxima.

A \textbf{piecewise function} is one that has different expressions based on different intervals of the independent variable. In simpler terms, the rules of the function change depending on the input value. This is often the case when the function includes an absolute value, as with the provided exercise.

An absolute value function like \( y = 3 - |t-3| \) is inherently piecewise because the absolute value expression \( |t-3| \) will have one form when the quantity inside is positive and another form when it's negative. In the exercise, for \( t < 3 \) the function behaves like \( y = 3 - (3 - t) \) and for \( t \geq 3 \) it behaves like \( y = 3 - (t - 3) \) . This affects how we calculate the rates of change and identify the critical points, which ultimately leads us to determine the absolute extrema.

The \textbf{absolute value} of a number is its distance from zero on a number line, without considering direction. It is always non-negative. When dealing with functions, the presence of an absolute value expression can cause the function to have critical points where the derivative does not exist or is not continuous.

In our function \( y = 3 - |t-3| \) , the absolute value affects the output of the function based on whether the argument \(t-3\) is positive or negative, resulting in a 'V' shaped graph. It's the kink at the vertex of this 'V' — in this case, at \( t = 3 \) — where the function is not differentiable, and hence a critical point is formed. Understanding the graph of the absolute value function helps when trying to find the highest and lowest points, which are the absolute extrema.