UPSEE Solved Model Paper 7

Given :
f (x) =

x

4+x+x2

, x ∊ [- 1 , 1]
We have to find the maximum value of f (x)
Now,
Differentiating f (x) w.r.t x, we get
f'(x) =

(4x+x2)−(2x+1)x

(4+x+x2)2

=

4+x+x2−2x2−x

(4+x+x2)2

=

4−x2

(4+x+x2)2

⇒ f'(x) > 0 for all x ∊ (- 2 , 2)
⇒ In the interval (-2, 2), f (x) is increasing
⇒ f (x) is also increasing in [-1 , 1] ⊂ (-2 , 2)
⇒ f (x) is maximum at x = 1
∴ fmax = f (1) =

1

4+1+12

=

1

4+1+1

= (

1

6

)
Hence, option 'D' is correct.