JEE Mains Model Paper 15

f'(x) =

1

8x

- b + 2x ; f''(x) = −

1

8x2

+ 2
For max or min , f'(x) = 0
1 - 8bx + 16x2 = 0
∴ x =

8b±√64b2−64

32

=

b±√b2−1

4

Case I. When b = 0 , f' (x) =

1

8x

+ 2x ≠ 0
Hence there is no extreme value for b = 0
Case II. When 0 < b < 1 , then b2 - 1 < 0
∴ x is not real
∴ there is no extreme value for 0 < b < 1
Case III. When b = 1 , then x =

1

4

f''(x) = 2 - 2 = 0
f''' (x) =

1

4x3

= 16 ( ≠ 0 ) for x =

1

4

∴ x =

1

4

is a point of inflexion
Case IV. When b > 1
Clearly α < β
We have
f'(x) =

1

8x

- b + 2x =

2

x

(x2−

1

2

bx+

1

16

) =

2

x

[(x−

b

4

)2−

1

16

(b2−1)]
=

2

x

[(x−

b

4

−

1

4

√b2−1) (x−

b

4

+

1

4

√b2−1)]
=

2

x

(x - β) (x - α) =

2

x

(x - α) (x - β)
∴ f'(x) > 0 , for 0 < x < α
f'(x) < 0 , for α < x < β and f'(x) > 0 for x > β
∴ f(x) has a local max at
x = α =

1

4

(b - √b2−1) and local min at
x = β =

1

4

(b + √b2−1)