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JEE Advanced2013MathematicsApplication of DerivativesActual

Paragraph: Let f:[0,1] R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies f^ (x)-2 f^ (x)+f(x) e^ x , x [0,1] . Question: Which of the following is true for f(x) ?

Options

  1. A0 < f ⁡ x < ∞
  2. B- 1 2 < f ⁡ x < 1 2
  3. C- 1 4 < f ⁡ x < 1
  4. D- ∞ < f ⁡ x < 0

Correct answer

D. - ∞ < f ⁡ x < 0

Step-by-step solution

f ″ x - 2 f ′ x + f x ≥ e x f ″ x · e - x - f ′ x e - x - f ′ x e - x + f x e - x ≥ 1 d dx f ′ x e - x - d dx f x · e - x ≥ 1 d dx f ′ x e - x - f x e - x ≥ 1 ⇒ d 2 dx 2 e - x f x ≥ 1 ∀ x ∈ 0 ,  1 Let ϕ x = e - x f x ⇒ ϕ x is concave upward f ( 0 ) = f ( 1 ) = 0 ⇒ ϕ 0 = 0 = ϕ 1 ⇒ f x < 0 ⇒ ϕ x < 0

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