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TIFR-2014-Maths-A-6

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Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ be a continuous function. Which of the following statements is always true? 

  1. $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{1} f(x) \text{d}x)^{2}$
  2. $\int_{0}^{1} f^{2}(x) \text{d}x \leq (\int_{0}^{1}| f(x) |\text{d}x)^{2}$
  3. $\int_{0}^{1} f^{2}(x) \text{d}x \geq (\int_{0}^{1}| f(x) |\text{d}x)^{2}$
  4. $\int_{0}^{1} f^{2}(x) \text{d}x ≰ (\int_{0}^{1} f(x) \text{d}x)^{2}$
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