Let f be a continuous function defined on [0, 1] such that \int^{1}_{0}f^2(x)dx=(\int^{1}_{0}f(x)dx)^2. Then the range of f

Correct Answer is : is a singleton

KVPY SX SB Exam 2016 Question Paper

Question : 88 of 120

Marks: +1, -0

Let f be a continuous function defined on [0, 1] such that

∫↖{1}↙{0}f^2(x)dx=(∫↖{1}↙{0}f(x)dx)^2

. Then the range of f

Solution:

By Cauchy Schwarz inequality

(∫↖{b}↙{a}f(x)g(x)dx)^2

≤ ∫↖{b}↙{a}(f(x))^2dx∫↖{b}↙{a}(g(x))^2dx

Here g(x) = 1
and equality holds only when

{f(x)}/{g(x)} = λ

So, f(x) is constant

Go to Question:

Prev Question

Next Question