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JEE Advanced 2013Paper 2
Exam
JEE Advanced
Year
2013
Paper
Paper 2
Subject
Mathematics
Chapter
Application of Derivatives

JEE Advanced 2013 · Paper 2 · Mathematics · Application of Derivatives

Paragraph: Let f:[0,1] arrow ℝ (the set of all real numbers) be a...

Paragraph: Let f:[0,1]→Rf:[0,1] \rightarrow \mathbb{R} (the set of all real numbers) be a function. Suppose the function ff is twice differentiable, f(0)=f(1)=0f(0)=f(1)=0 and satisfies f′′(x)−2f′(x)+f(x)≥ex,x∈[0,1].f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .
Question: If the function e−xf(x)\mathrm{e}^{-x} f(x) assumes its minimum in the interval [0,1][0,1] at x=14x=\frac{1}{4}, which of the following is true?

Options

  1. Af′(x)<f(x),14<x<34{f }^{′} \left(\text{x}\right) < f \left(\text{x}\right) \text{,} \dfrac{1}{4} < \text{x} < \dfrac{3}{4}
  2. Bf′(x)>f(x),0<x<14{f }^{′} \left(\text{x}\right) > f \left(\text{x}\right) \text{,} 0 < \text{x} < \dfrac{1}{4}
  3. Cf′(x)<f(x),0<x<14{f }^{′} \left(\text{x}\right) < f \left(\text{x}\right) \text{,} 0 < \text{x} < \dfrac{1}{4}
  4. Df′(x)<f(x),34<x<1{f }^{′} \left(\text{x}\right) < f \left(\text{x}\right) \text{,} \dfrac{3}{4} < \text{x} < 1

Correct answer

C. f′(x)<f(x),0<x<14{f }^{′} \left(\text{x}\right) < f \left(\text{x}\right) \text{,} 0 < \text{x} < \dfrac{1}{4}

Step-by-step solution

ϕ′(x)<0,x∈(0,1/4){ϕ}^{′} \left(\text{x}\right) < 0 \text{,} \text{x} \in \left(0 \text{,} 1 / 4\right) and ϕ′(x)>0,x∈(1/4,1)⇒e−xf′(x)−e−xf(x)<0,x∈(0,1/4){ϕ}^{′} \left(\text{x}\right) > 0 \text{,} \text{x} \in \left(1 / 4 \text{,} 1\right) ⇒ {\text{e}}^{- \text{x}} {\text{f}}^{′} \left(\text{x}\right) - {\text{e}}^{- \text{x}} \text{f} \left(\text{x}\right) < 0 \text{,} \text{x} \in \left(0 \text{,} 1 / 4\right)
f′(x)<f(x),0<x<1/4{\text{f}}^{′} \left(\text{x}\right) < \text{f} \left(\text{x}\right) \text{,} 0 < \text{x} < 1 / 4
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