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Question Number 202605 by hardmath last updated on 30/Dec/23

Commented by mr W last updated on 30/Dec/23

$w h a t d o y o u m e a n w h e n y o u w r i t e$ $\underset{= b}{m a x} \underset{= a}{m i n} (. . .) ?$
Commented by hardmath last updated on 30/Dec/23

$dear professor,$ In the given expression, the maximum value is for b, and the minimum value is for a
Commented by mr W last updated on 30/Dec/23

$x = \pm 1$
	Answered by mr W last updated on 30/Dec/23

	$w e r e p l a c e a, b, x w i t h x, y, k .$ $f (x, y) = x^{2} - 2 x y - y^{2} - 2 k x + 10 k y$ $= {(x - y - k)}^{2} - 2 y^{2} + 8 k y - k^{2}$ $f_{m i n} i s w h e n x - y - k = 0$ $f (x, y) = - {(y + x - 5 k)}^{2} + 2 x^{2} - 12 k x + 25 k^{2}$ $f_{m a x} i s w h e n x + y - 5 k = 0$ $\Rightarrow x = 3 k, y = 2 k$ ${(3 k)}^{2} - 2 (3 k) (2 k) - {(2 k)}^{2} - 2 k (3 k) + 10 k (2 k) = 7$ $k^{2} = 1$ $\Rightarrow k = \pm 1$
	Commented by hardmath last updated on 30/Dec/23

	$cool dear professor thank you$
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