The critical point and nature for the function f ( x , y ) = x 2 - 2 x + y 2 + 2 y - 2 is:

Question

The critical point and nature for the function $f (x, y) = x^{2} - 2 x + y^{2} + 2 y - 2$ is:

Smart Achievers · Accepted Answer

(4)

Ans. (1, - 1) min.

Explanation:

$f (x, y) = x^{2} - 2 x + y^{2} + 2 y - 2$

Partial Derivatives

$f' (x) = 2 x - 2$

and $f' (y) = 2 y + 2$

Therefore, for critical points,

$f' (x) = 0$

$2 x - 2 = 0$

$\Rightarrow$ $x = 1$

and $f' (y) = 0$

$2 y + 2 = 0$

$\Rightarrow$ $y = - 1$

So, the critical point is $(1, - 1) .$

Now, $f'' (x) = 2 > 0$ and $f'' (y) = 2 > 0$

So at $(1, - 1) .$

$(f_{x}^{''} \times f_{y}^{''}) - f_{x}^{'} f_{y}^{'} = 2 \times 2 - 0 = 4 > 0$

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