Q58 / 90 · KVPY 2016 (SB/SX) · Mathematics

Let $f$ be a continuous function defined on $[0,1]$ such that $\int_{0}^{1} f^{2}(x) d x=\left(\int_{0}^{1} f(x) d x\right)^{2}$. Then the range of $f$

  • A
    has exactly two points
  • B
    is the interval $[0,1]$
  • C
    has more than two points
  • Correct answer
    D
    is a singleton

Solution

By Cauchy Schwarz inequality $$ \left\{\int_{a}^{b} f(x) g(x) d x\right\}^{2} \leq \int_{a}^{b}(f(x))^{2} d x \int_{a}^{b}(g(x))^{2} d x $$ Here $g(x)=1$ and equality holds only when $\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\lambda$ So, $f(x)$ is constant

Hence, the correct answer is option: 'D'.

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