Question

Let f be a continuous function defined on $[0,1]$ such that ${\int }_0^1{f}^2\left(x\right)dx={\left({\int }_0^{1}f\left(x\right)dx\right)}^2$. Then the range of f.

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

Answer 1

The correct option is C is the interval $[0,1]$

consider constant function $y=K$

Verify it :

$LHS={\int }_0^1{K}^{2}dx=K^2$

$RHS={\left({\int }_0^{1}Kdx\right)}^2=K^2$

Hence answer is C.