MinMax Secant Lines


The function shown is
It has extrema values at . The applet shows secant lines as the distance between points is varied. The derivative is the limit as if both left and right limits are the same, or one of the limits is outside of the domain. The points where extrema may exist are where the derivative is zero or undefined.
Place the orange plus symbol on each of the extrema values one at a time. Vary by moving the slider or turning on animation. Note whether the left and right secant line slopes approach the same value as gets small in the right graph. Turning on the trace will show the limit relation better. Comment on whether an extrema would not exist where the derivative does not exist ( different left and right slope limits, infinite slope limits.