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KVPY2020MathematicsApplication of Derivatives

Let f ( x ) = sin x + x 3 - 3 x 2 + 4 x - 2 cos x for x ∈ ( 0 , 1 ) . Consider the following statements I. f has a zero in 0 ,   1 II. f is monotone in 0 ,   1 Then

Options

  1. AI and II are true
  2. BI is true and II are false
  3. CI is false and II are true
  4. DI and II are false

Correct answer

A. I and II are true

Step-by-step solution

f x = sin x + x 3 - 3 x 2 + 4 x - 2 cos x , x ∈ ( 0 , 1 ) x ∈ ( 0 , 1 ) f ( 0 ) = - 2 > 0 f ( 1 ) = sin 1 < 0 ∵ f ( 0 ) · f ( 1 ) < 0 ⇒ f ( x ) has a zero in 0 ,   1 Now, f x = sin x + ( x - 1 ) 3 + ( x - 1 ) cos x ⇒ f ' x = 3 ( x - 1 ) 2 + 2 cos x - sin x ( x - 1 ) 3 + x - 1 = 3 ( x - 1 ) 2 + 2 cos x + ( 1 - x ) 3 + ( 1 - x ) sin x > 0 ∀ x ∈ ( 0 , 1 ) ⇒ f ( x ) is monotone in 0 ,   1

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