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EEJ MAIN Mathematics QUESTION #6030
Question 1

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable with $|f(x)-f(y)|\le 2|x-y|^{3/2}$ for all $x,y\in\mathbb{R}$. If $f(0)=1$, evaluate $\displaystyle\int_0^1 f^2(x)\,dx$.

  • 1✔️
  • 2
  • $\dfrac{1}{2}$
Correct Answer Explanation

From the condition $|f(x)-f(y)|\le 2|x-y|^{3/2}$, divide both sides by $|x-y|$:

$\left|\frac{f(x)-f(y)}{x-y}\right|\le 2|x-y|^{1/2}$

Taking $y\to x$: $|f'(x)|\le 2\cdot0=0$, so $f'(x)=0$ for all $x$.

Hence $f$ is constant. Since $f(0)=1$, we have $f(x)=1$ for all $x$.

$\int_0^1 f^2(x)\,dx = \int_0^1 1\,dx = \mathbf{1}$