math104-s21:hw12
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math104-s21:hw12 [2021/04/24 05:07] pzhou |
math104-s21:hw12 [2026/02/21 14:41] (current) |
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| Hint: | Hint: | ||
| (a) If $u \geq 0, v \geq 0$, then $$ uv \leq \frac{u^p}{p} + \frac{v^q}{q} $$ | (a) If $u \geq 0, v \geq 0$, then $$ uv \leq \frac{u^p}{p} + \frac{v^q}{q} $$ | ||
| + | If you cannot prove this, you may assume it and proceed (no points taken off). If you want to prove it, you may fix $u$ and let $v$ vary from $0$ to $\infty$, and watch how $\frac{u^p}{p} + \frac{v^q}{q} - uv$ change, and obtain that at the minimum the quantity is still non-negative. | ||
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| (b) If $f, g$ are **non-negative** Riemann integrable functions on $[a,b]$, and | (b) If $f, g$ are **non-negative** Riemann integrable functions on $[a,b]$, and | ||
| $$ \int f^p dx = 1, \quad \int g^q(x) dx = 1 $$ | $$ \int f^p dx = 1, \quad \int g^q(x) dx = 1 $$ | ||
math104-s21/hw12.1619240879.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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