Paragraph: Let $f:[0,1] \rightarrow \mathbb{R}$ (the set of all real numbers) be a function. Suppose the…

Paragraph: Let $f:[0,1] \rightarrow \mathbb{R}$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .$ Question: Which of the following is true for $0 < x < 1$ ?
  1. 0 < f⁡ x < ∞
  2. - 1 2 < f⁡ x < 1 2
  3. - 1 4 < f⁡ x < 1
  4. - ∞ < f⁡ x < 0

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

Asked in: JEE Advanced 2013 (Paper 2)

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