By Seán Dineen (auth.)
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Extra resources for Functions of Two Variables
Example text
5 ohms. Find an upper bound for your error. 8 Tangent Planes Summary. se at a non-critical point and are led to the problem of finding the maximum of a function on a level set. We solve this in the next chapter using Lagrange multipliers. We consider the geometric analogue of linear approximation and define the tangent plane. Until now we have concentrated on the behaviour of a function near a critical point. We now consider functions near a non-critical point. sing near x and f'(x) < 0 implies that f is decreasing near x.
1). 1 These examples show that if we have too many derivatives vanishing at Functions 01 two variables 30 a critical point then we can say very little. b)) = 0 at a critical point (a,b) to be the analogue of the condition f"(a) = 0 for a function of one variable. We call such points degenerate critical points and it is necessary to use other methods, which we will not discuss, to decide if a degenerate critical point is a local maximum or minimum or neither. If det(H,) =I- 0 at the critical point (a,b) then we call the critical point a non-degenerate critical point.
Proposition 25. If I is defined on an open subset ofR2 and at (a, b) we have '\l I(a, b) = (0,0) and det(Hf(a,b)) > 0 then (a) (a, b) is a local maximum if ~:; (a, b) < 0 (b) (a, b) is a local minimum if ~:~ (a, b) > O. So far, so good. We have found a method for locating local maxima and local minima. Are there any other possibilities? 2 Find the critical points of the following functions (a) x 2 + 2xy + 3y2 (b) 4x 3 + 4xy - y2 - 4x (e) sinxsinysin(x+y) (f) xY+X-1+y-l. 3 A function ~:; + ~:; 1 defined on an open set in R2 is said to be harmonic if = 0 at all points.