Definite Integration
Cauchy-Schwarz optimization — paragraph I
MJAT_TS4_P2
Grade 12
Question:
**Paragraph I:**
Let $f(x)$ be a continuous function over $[0,1]\to\mathbb{R}$ such that $\displaystyle\int_0^1 f^2(x)\,dx=1$ and $L=\displaystyle\int_0^1 x^{2021}f(x)\,dx$.
**Question:** The maximum value of $L$ is $\dfrac{1}{\sqrt{\alpha}}$ where $\alpha=$
Step-by-Step Solution
Key Concept: By Cauchy-Schwarz: $L^2\leq\left(\int_0^1 x^{4042}\,dx\right)\left(\int_0^1 f^2(x)\,dx\right)=\frac{1}{4043}\cdot 1$. So $\max L=\frac{1}{\sqrt{4043}}$.
$\alpha=\mathbf{4043}$.
Correct Answer: 4043