We know that a function f is increasing where f ' > 0 and decreasing where f ' < 0. This is easy to implement on the TI-89. For instance, is y = x3 - 3x + 5 increasing at x = 3? Type "d(x3 - 3x + 5, x)|x=3" (You can get the derivative function from the menu, or press ) and press . If the result is greater than zero, the answer is "yes". To test other values of x, just erase the "3", type the new value, and press .
This process can be made even easier by means of a user-defined function. You could use either:
"Define isincr(f, x, c) = (d(f, x)|x=c)>0" for a user-defined function to test for increasing, or:
"Define isdecr(f, x, c) = (d(f, x)|x=c)< 0" to test for decreasing.
Either function returns a "true" or "false" result. You really don't need both functions, since if the derivative isn't zero or positive, it must be negative, right? You can use one of these functions in conjunction with the critnum() function to implement a very fast and convenient First Derivative Test. First, use the critnum() function to locate critical numbers (By the way, don't forget that the critnum() function does not locate critical numbers where f ' is undefined - that remains your job. Poor you!), then either isincr() or isdecr() to test for increasing/decreasing in each interval. Slick!