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Let a function f : R ® R be defined as:

  f ( x ) = { ∫ 0 x ( 5 − | t − 3 | ) d t , x > 4 x 2 + b x                                     , x ≤ 4           

where b ∈ R .  If f is continuous at x = 4, then which of the following statements is NOT true?

Option 1 -

f if not differentiable at x = 4

Option 2 -

f'(3) + f'(5) =   3 5 4

Option 3 -

f is increasing in ( − ∞ , 1 8 ) ∪ ( 8 , ∞ )

Option 4 -

f has a local minima at   x = 1 8

0 2 Views | Posted 2 months ago
Asked by Shiksha User

  • 1 Answer

  • A

    Answered by

    alok kumar singh | Contributor-Level 10

    2 months ago
    Correct Option - 3


    Detailed Solution:

      f ( x ) { ∫ 0 3 ( 2 − t ) d t + ∫ 3 x ( 8 − t ) d t ; x > 4 x 2 + b x                           ; x ≤ 4

    f(x) is continuous at x = 4

    1 6 + 4 b = ∫ 0 3 ( 2 − t ) d t + ∫ 3 4 ( 8 − t ) d t           

    16 + 4b = 15

    ⇒ b = − 1 4               

    f o r     x ≤ 4 , f ( x ) = x 2 − x 4 f ' ( x ) = 2 x − 1 4                 

    f(x) is increasing in  ( 1 8 , ∞ )  

     rate change at x = 1 8 ,  from -ve to +ve. So minima occurs.

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f (x) is an even function

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So, f (x) has at least four roots in (-2, 2)

g ( − 3 4 ) = g ( 3 4 ) = 0         

So, g (x) has at least two roots in (-2, 2)

now number of roots of f (x) ⋅ g " ( x ) = f ' ( x ) ⋅ g ' ( x ) = 0  

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Vishal Baghel

f ( x ) = x + x ∫ 0 1 f ( t ) d t − ∫ 0 1 t 0 f ( t ) d t

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Now, α = ∫ 0 1 f ( t ) d t + 1

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A
alok kumar singh

Kindly go through the solution 

 

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