Q51 / 89 · KVPY 2017 (5 Nov SB/SX) · Mathematics

The number of continuous functions $\mathrm{f}:[0,1] \rightarrow \mathrm{R}$ that satisfy

  • A
    0
  • Correct answer
    B
    1
  • C
    2
  • D
    infinity

Solution

Given equation can be written as $-\frac{1}{3}=\int_{0}^{1}\left(\frac{\mathrm{f}(\mathrm{x})}{2}-\mathrm{x}\right)^{2} \mathrm{dx}-\int_{0}^{1} \mathrm{x}^{2} \mathrm{dx}$ $$ \begin{array}{l} \int_{0}^{1} x^{2} d x \frac{1}{3}=\int_{0}^{1}\left(\frac{f(x)}{2}-x\right)^{2} d x \\ 0=\int_{0}^{1}\left(\frac{f(x)}{2}-x\right)^{2} d x \end{array} $$ So, no of continuous function is 1 .

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

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