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KVPY2016MathematicsDefinite Integration

Let f be a continuous function defined on [0,1] such that ₀¹ f²(x) d x= ( ₀¹ f(x) d x )² . Then the range of f

Options

  1. Ahas exactly two points
  2. Bis the interval [0,1]
  3. Chas more than two points
  4. Dis a singleton

Correct answer

D. is a singleton

Step-by-step solution

By Cauchy Schwarz inequality _ a ^ b f(x) g(x) d x ² _ a ^ b (f(x))² d x _ a ^ b (g(x))² d x Here g(x)=1 and equality holds only when f ( x ) g ( x ) = So, f(x) is constant

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