Find the local extreme values of the function : f(x,y)= xy−x^2 −y^2 −2x−2y+4.


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Question Number 69162 by Learner-123 last updated on 20/Sep/19

$F i n d t h e l o c a l e x t r e m e v a l u e s o f$ $t h e f u n c t i o n :$ $f (x, y) = x y - x^{2} - y^{2} - 2 x - 2 y + 4 .$
	Answered by MJS last updated on 20/Sep/19

	$\frac{d}{d x} [- x^{2} - y^{2} + x y - 2 x - 2 y + 4] = - 2 x + y - 2$ $- 2 x + y - 2 = 0 \Rightarrow x = \frac{y - 2}{2}$ $f (\frac{y - 2}{2}, y) = - \frac{3}{4} y^{2} - 3 y + 5$ $\frac{d}{d y} [- \frac{3}{4} y^{2} - 3 y + 5] = - \frac{3}{2} y - 3$ $- \frac{3}{2} y - 3 = 0 \Rightarrow y = - 2 \Rightarrow x = - 2$ $the 2^{nd} derivates are < 0 \Rightarrow maximum at$ $(\begin{matrix} x \\ y \\ f (x, y) \end{matrix}) = (\begin{matrix} - 2 \\ - 2 \\ 8 \end{matrix})$
	Commented by Learner-123 last updated on 23/Sep/19

	$t h a n k s s i r .$
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