Q49 / 90 · KVPY 2016 (SB/SX) · Mathematics

The number of continuous function $f:[0,1] \rightarrow[0,1]$ such that $\mathrm{f}(\mathrm{x}) < \mathrm{x}^{2}$ for all $\mathrm{x}$ and $\int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{1}{3}$ is :

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

Solution

$\therefore \mathrm{f}(\mathrm{x})$ is always positive for $\mathrm{x} \in[0,1]$ $\begin{array}{c} \therefore \mathrm{f}(\mathrm{x}) < \mathrm{x}^{2} \Rightarrow \int_{0}^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} < \int_{0}^{1} \mathrm{x}^{2} \\ \mathrm{I} < \frac{1}{3} \end{array}$ But it is given that $\mathrm{I}=\frac{1}{3}$ which is not possible

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

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