Maximum of a Function

Suppose we want to compute the relative(local) maximum for the function .

Here is the graph of this polynomial. For this graph I used X-min = - 5, X-max = 7, Y-min = -75, and Y-Max = 75

Method 1: Using the command fMax

They syntax for this command is: fMax(function,X,left bound, right bound)

This command will return the x-value of the point where the function has a maximum.

WARNING: This command is equivalent to asking where the absolute maximum is located on the given interval. Randomly picking the left bound and right bound will tend to give you an incorrect answer.

From the home screen do the following.

• Press
• Choose 7:fMax(
• Now enter the function.
  • Sometimes it is easier to enter the function in the screen and then use a variable name.
• Press
• Type the left bound. This is some x-value that is smaller than where the actual max is located.
• Press
• Type the right bound. This is some x-value that is larger than where the actual max is located.
• Press
• Press

Using the example from above with the interval x=0 to x=3 should now display the following. Note: The interval was found using the graph of the function given above.

Now to find the actual maximum value, evaluate the function at x=1.

Method 2: Using a Graphing Screen

• Press and enter the function into Y₁.
  • Make sure the function is turned on.
• Set the window. You might have to play with the window settings until you get a good window. For this problem I used

X-min = - 5, X-max = 7, Y-min = - 75, and Y-Max = 75

• Once you have your graph, press
• Choose 4:maximum.
• You will now see the words LeftBound?. Enter the value for the left bound or move the curser to the left of where the max occurs. will move the cursor quickly to the left.

This is what you should see. I used x=0 for the left bound.

• Press
• You will now see the words RightBound?. Enter the value for the right bound or move the curser to the right of where the max occurs. will move the cursor quickly to the right.

This is what you should see. I used x=3 for the left bound.

• Press
• Now the calculator will ask for a Guess. Move the cursor close to where the maximum occurs and press .

The calculator should now display the following.

Thus the relative maximum for the function is 17 and it occurs at x=1.