We know that a function f is concave up where f " > 0 and concave down where f " < 0. This is easy to implement on the TI-89. For instance, is y = x3 - 3x + 5 concave up or down at x = 3? Type "d(x3 - 3x + 5, x, 2)|x=3" (You can get the derivative function from the menu, or press ) and press . If the result is positive, the answer is "concave up", and if the answer is negative, the answer is "concave down" . 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 isconup(f, x, c) = (d(f, x,2)|x=c)>0" for a user-defined function to test for concave up, or:
"Define iscondn(f, x, c) = (d(f, x, 2)|x=c)< 0" to test for concave down.
Either function returns a "true" or "false" result. You really don't need both functions, since if the second derivative isn't zero or positive, it must be negative, right? You can use one of these functions in conjunction with the ptinf() function to implement a very fast and convenient test for intervals of concavity. To implement the Second Derivative Test, it would be probably more convenient to use the critnum() and ismax() and ismin() functions, although you could use the critnum(), ptinf(), and isconup() functions as follows. 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!). Secondly, use the ptinf() function to be sure that your critical numbers are points of inflection. Then either isincr() or isdecr() will test each remaining critical number for maxima/minima. Slick!