math104-s21:hw12
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math104-s21:hw12 [2021/04/23 22:06] pzhou |
math104-s21:hw12 [2022/01/11 10:57] (current) pzhou ↷ Page moved from math104-2021sp:hw12 to math104-s21:hw12 |
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| ====== HW 12 ====== | ====== HW 12 ====== | ||
| - | 1. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \In [a,b]$, $f$ is integrable on $[c, | + | 1. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \subset |
| 2. (2 point) If $f$ is a continuous non-negative function on $[a,b]$, and $\int_a^b f dx = 0$, then $f(x)=0$ for all $x \in [a, | 2. (2 point) If $f$ is a continuous non-negative function on $[a,b]$, and $\int_a^b f dx = 0$, then $f(x)=0$ for all $x \in [a, | ||
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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/hw12.1619240812.txt.gz · Last modified: 2021/04/23 22:06 by pzhou
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