# Finding Minimums and Maximums

A relative minimum is a point that is lower than all the other points around it. A relative maximum is a point that is higher than all the other points around it. Sure, there are other, more precise, definitions, but that will work for what we want to do. The generic word for minimum or maximum is extremum.

## All calculators

1. Solve the equation for Y if it isn't already that way.
2. Go to Y=
3. Enter the function
4. Hit Graph
5. Change the viewing window if necessary so that you can see where the graph has a minimum or maximum. You need to make sure that there is enough to the left and right of the extremum to select a point.
6. Continue on to the specific steps for your calculator. If there are additional extrema to find, you may need to change your viewing window and repeat the steps for each extremum.

## TI-82

1. Press Calc (2nd Trace)
2. Choose Minimum (#3) or Maximum (#4)
3. For the Lower Bound, arrow to the left of the extremum and press enter.
4. For the Upper Bound, arrow to the right of the extremum and press enter.
5. For the Guess, arrow to the extremum and press enter.
6. The TI-82 will return a value for x and y. The x is where the extremum occurs and the y is the minimum or maximum value.

## TI-83

For the TI-83, you may enter left bounds, right bounds, and guesses from the keyboard instead of arrowing and hitting enter. When it asks for a value, just enter your number and press enter.

1. Press Calc (2nd Trace)
2. Choose Minimum (#3) or Maximum (#4)
3. For the Left Bound, arrow to the left of the extremum and press enter.
4. For the Right Bound, arrow to the right of the extremum and press enter.
5. For the Guess, arrow to the extremum and press enter.
6. The TI-83 will return a value for x and y. The x is where the extremum occurs and the y is the minimum or maximum value.

## TI-85 / TI-86

1. Press Graph
2. Press More and then Math (F1)
3. Press Lower (F1). Arrow to the left of the extremum and press enter.
4. Press Upper (F2). Arrow to the right of the extremum and press enter.
5. Press More and then FMin (F1) for minimum or FMax (F2) for maximum.
6. Arrow to the extremum and press enter.
7. The TI-85 will return a value for x and y. The x is where the extremum occurs and the y is the minimum or maximum value.

## TI-89 / TI-92

For the TI-89 and TI-92, you may enter lower bounds or upper bounds from the keyboard instead of arrowing and hitting enter. When it asks for a value, just enter your number and press enter.

1. Press Math (F5)
2. Choose Minimum (#3) or Maximum (#4)
3. For the Lower Bound, arrow to the left of the extremum and press enter.
4. For the Upper Bound, arrow to the right of the extremum and press enter.
5. The TI-89 will return a value for x and y. The x is where the extremum occurs and the y is the minimum or maximum value.

You do not need to supply a guess for the TI-89. You should make sure there is only one extremum in the interval specified or you won't know which one it is going to give you.

## Example (TI-83+)

Find the relative maximum for y = x3 - 2x2 - 4x

Enter the function into y1 and then graph. You may need to change your viewing window so that the maximum is visible. I've used a zoom decimal for this example. I've also entered the left bound, right bound, and guess from the keyboard rather than arrowing and pressing enter.

Graph Zero Left Right Guess Solution

The calculator says the solution is x = -0.6666651 and y = 1.4814815.

If you wanted to find the relative minimum, you would need to change your viewing window. Allow a little extra space below the minimum so that the text for the answer doesn't overwrite your graph.

For this problem, the window settings: XMin = -1, XMax = 4, YMin = -10, and YMax = 3 work fairly well. The image below shows the graph with those window settings.

Window Graph

You can find the minimum on your own. If you would like to know if you have the right answer, it is x = 2 and y = -8.

Note that you do not have to have the maximum showing to find the minimum. This is true of all features of the calc menu. Only the part of the graph that you're interested in needs to be visible. Other x-intercepts, maximums, minimums, or intercepts don't have to be showing.