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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. $\int_{0}^{1} f^{2}(x) \text{d}x ≰ (\int_{0}^{1} f(x) \text{d}x)^{2}$
asked in Calculus | 58 views