One of the most important applications of the ideas of calculus is to optimization: trying to find the maximum or minimum values of some function that we are interested in. In this lab we will explore how to find these extreme values and some applications.

To find the extreme values of a function, we need to think about what happens to the function at these points. Recall the following definitions from the lectures.

We say a function \( f \) has an absolute maximum on an interval \( [a,b] \) at \( x=c \) if \( f(c)\geq f(x)\) for all \( a \leq x \leq b \); we say a function \( f \) has an absolute minimum at \( x=c \) if \( f(c) \leq f(x)\) for all \( a \leq x \leq b \). If \( f \) has either an absolute maximum or an absolute minimum on \( [a,b] \) at \( x=c \), we say that \( f \) has an absolute extremum at \( x=c \).

On the other hand, we say a function \( f \) has a local maximum at \( x=c \) if \( f(c) \geq f(x)\) for all \( x \) near \( c \) (though there may be larger values further away from \( x=c \)); we say a function \( f \) has a local minimum at \( x=c \) if \( f(c) \leq f(x) \) for all \( x \) near \( c \). Finally, \( f \) has a local extremum at \( x=c\) if it has either a local maximum or a local minimum at that point.

We say that \( x = c \) is a critical point of a function \( f \) if either \( f'(c) = 0 \) or \( f'(c) \) does not exist.

The next plot shows both the graph of a function \( f(x) \) on the left and its derivative \( f'(x) \) on the right. A slider bar allows you to move a point along the graph of each function.

1. Explore the graph using the slider bar and use it to answer the following questions in your lab report.