Questions Asked in Previous 10 Years' Exams - Directions Let \( f:[0,1] \rightarrow R \) (the se...

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Questions Asked in Previous 10 Years' Exams
- Directions Let \( f:[0,1] \rightarrow R \) (the set of all real numbers) be a function. Suppose the function \( f \) is twice differentiable, \( f(0)=f(1)=0 \) and satisfies
\[
f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1]
\]
[Passage Based Questions 2013]
If the function \( e^{-x} f(x) \) assumes its minimum in interval \( [0,1] \) at \( x=1 / 4 \), then which of the following is true?
\( \begin{array}{ll}\text { (c) } f^{\prime}(x)f(x), 0x\frac{1}{4} & \text { (d) } f^{\prime}(x)f(x), \frac{3}{4}x1\end{array} \)
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