 CALCULUS Understanding Its Concepts and Methods

Tests for extreme values

The function f has a local maximum at a point c if f(c) ≥ f(x) for every point x that is sufficiently near c.

The function f has a local minimum at a point c if f(c) ≤ f(x) for every point x that is sufficiently near c.

Second derivative testLet f be a function such that f'(c) = 0 and the second derivative exists for x near c. If f''(c) < 0, then f(c) is a local maximum. If f''(c) > 0, then f(c) is a local minimum. If f''(c) = 0, then the test fails.

The function f(x,y) has a local maximum at (a,b)  if f(a,b) ≥ f(x,y)  for all points (x,y) sufficiently close to (a,b).

The function f(x,y) has a local minimum at (a,b)  if f(a,b) ≤ f(x,y)  for all points (x,y) sufficiently close to (a,b).

Test for extremaAssume f(x,y) has continuous second partial derivatives at  (a,b) and suppose fx(x,y) = 0  and fy(x,y) = 0.

Let D = fxx(a,b)fyy(a,b) - (fxy(a,b))2 . Then at (a,b), the function f has a local maximum if D > 0 and fxx(a,b) < 0; a local minimum if D > 0 and fxx(a,b) > 0; a saddle point if D < 0.

Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.