Let f be a continuous function defined on [0,1] such that _0^1 f^… - Vidyadip

MCQ

Let $f$ be a continuous function defined on $[0,1]$ such that $\int_0^1 f^2(x) d x=\left(\int_0^1 f(x) d x\right)^2$. Then, the range of $f$

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

✓

Answer

Correct option: D.

is a singleton

d
(d)

We have,

$\int \limits_0^1 f^2(x) d x=\left(\int \limits_0^1 f(x) d x\right)^2, x \in[0,1]$

We know Cauchy Schwartz inequality

${\left[\int_a^b f(x) \cdot g(x) d x\right]^2 \leq \int_a^b(f(x))^2 d x }$

$\int_a^b(g(x))^2 d x$

Here, $g(x)=1$ and equality holds only when $\frac{f(x)}{g(x)}=\lambda$

So, $f(x)$ is constant function.

$\therefore f(x)$ is a singleton.

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