The Second Derivative Test for Local Maxima and Minima says: a. has a local maximum value at x = c if (c) = 0

The Second Derivative Test for Local Maxima and Minima says:

a. ƒ has a local maximum value at x = c if ƒ′(c) = 0 and ƒ″(c)

b. ƒ has a local minimum value at x = c if ƒ′(c) = 0 and ƒ″(c) > 0.

To prove statement (a), let P = (1/2) |ƒ″(c)|. Then use the fact that

to conclude that for some δ > 0,

Thus, ƒ′(c + h) is positive for -δ

f"(c)= lim h0 f'(c + h) - f'(c) h lim f'(c + h) h