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A) If (a, b) is a critical point and det(HJ(a,b») > 0 then, at (a, b), f has a local maximum if ~:; (a, b) < 0 and a local minimum if ~:; (a, b) > O. (b) If (a, b) is a critical point and det(H/(a,b») < 0 then (a, b) is a saddle point. (c) If (a, b) is a critical point and det(H/(a,b») = 0 then (a, b) is a degenerate critical point and other methods are required in order to determine whether it is a local maximum, a local minimum or a saddle point. FUnctions of two variables 32 Example 27. Let f(x,y) = x 2 - 3xy - y2 - 2y - 6x.

OfR and 8x 2 ' 8y2' 8x8y and 8y8x all eXlst and are all contmuous then 82 1 82 1 8x8y = 8y8x' We will define continuous functions soon but will not prove this proposition. However, we will provide a method for recognizing many functions which satisfy the hypothesis of this proposition. This will not involve working out all the derivatives and so will be quite practical. 'IT . h functlOns . £,or wh'ICh 8x8y 8 21 = 8y8x' 8 21 ne now assume we are d eal'mg WIt These derivatives are called-for obvious reasons-the mixed second order partial derivatives.

Since e Y =/: 0, no matter what the value of y, we have arrived at something which is impossible and so we cannot have a solution in this case. 4). 4 Now 82 f 8 8x 2 = 8x (e zY (1 = 82 f and 8x 2 (0, 1) = 1eo(1 + + xy - y)) ye zY (1 + xy - y) + ye zy °- 1) + -88y2f = -8y8 ((x 2 82 f and 8y2 (0, 1) 82 f 82 f = (0 - 1) . eO. 1)xe zy ) °. °= eO . 1 = 1. Also = (x - 1)xe zy ; x 0 8 8x8y = 8y8x = 8y (e ZY (1 + xy - y)) = xe zY (1 + xy - y) + eZY(x - 1). 82 f 80 88- (O,1)=O·eO(1+0·1-1)+eO(O-1)=-1. x y Functions of two variables 34 We have Hf(O,l) = (_~ -~) and det (Hf(O,l») and (0,1) is a saddle point of the function = -1 < 0 f.

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Introduction to fourier series and integrals by Seeley


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