UPSC NDA Math Model Paper 6

Here, f(x,y)=x2−2x+2y2+4y−2
Partial derivatives,
f′(x)=2x−2 and f′(y)=4y+4
Now, for critical points, f′(x)=0
⇒2x−2=0
⇒x=1,
Also, f'(y) = 0
⇒ 4y + 4 = 0
⇒y = -1,
So, critical points (1, -1)
f''(x) = 2 > 0 and f''(y) = 4 > 0
So, at (1, -1) minimum
Hence, option (4) is correct.