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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: If the function e ^ -x f(x) assumes its minimum in the interval [0,1] at x= 1 4 , which of the following is true?

Options

  1. Af ⁡ ′ x < f ⁡ x , 1 4 < x < 3 4
  2. Bf ⁡ ′ x > f ⁡ x , 0 < x < 1 4
  3. Cf ⁡ ′ x < f ⁡ x , 0 < x < 1 4
  4. Df ⁡ ′ x < f ⁡ x , 3 4 < x < 1

Correct answer

C. f ⁡ ′ x < f ⁡ x , 0 < x < 1 4

Step-by-step solution

ϕ ′ x < 0 ,  x ∈ 0 ,  1 / 4 and ϕ ′ x > 0 ,  x ∈ 1 / 4 ,  1  ⇒  e - x f ′ x - e - x f x < 0 ,  x ∈ 0 ,  1 / 4 f ′ x < f x ,  0 < x < 1 / 4

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