If x ∈ R, then minimum and maximum limit of   are

The correct option is ( 3 ).

Concept : 

Maximum and Minimum Values of the Quadratic Expression :

Let's suppose y = ax² + bx + c ,

when determining the maximum and minimum values of ax² + bx + c.

suppose x is real ,then the descriminant of equation 

ax2 + bx + c - y = 0

⇒ b² - 4a(c - y) = 0

Calculation :

Let,

y = 

yx² + 2xy + 9y = x² - 2x + 9

x² ( y - 1 ) + x ( 2y + 2 ) + 9 ( y - 1 ) = 0

Now,

D = 0

b² - 4ac = 0

⇒ ( 2y + 2 )²  - 4( y - 1 ) 9 ( y - 1 ) = 0

4y² + 4 + 8y - 36y² - 36 + 72y = 0

32y² - 80y + 32 = 0

2y² - 5y + 2 = 0

2y ( y - 2 ) - 1 ( y - 2 ) = 0

( y - 2 ) ( 2y - 1 ) = 0

⇒ y = 2 and , 

Thus,

maximum value of the function = 2

minimum value of the function = 

Hence, The correct option is ( 3 ).