I have a problem with the minimum. Let:
$G(y)=\int_{y}^{a}g(x)dx$ , $g>0$ , $0\leq y< a$.
Is that the function $G(y)$ different to zero? Is that the function $G(y)$ admits a muinmum different to zero?
Minimum of a function defined by an integral
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integration
lebesgue-integral
1 Answers
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Notice that $G(a) = \int _a ^a g(x) \ \Bbb d x = 0$. Notice also that $G'(y) = -g(y) < 0$, which shows that $G : [0,a] \to \Bbb R$ is strictly decreasing, therefore $G(y) > G(a) = 0$ for all $0 \le y < a$, which shows that $G > 0$ on $[0,a)$. In particular, since $G$ is decreasing, it follows that its minimum is $G(a) = 0$.
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0Then, there is not a lower bound $m$ different to zero such that: $G(y)>m$ for all $0\leq y – 2017-01-12
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0@user326064: Exactly. – 2017-01-12
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0Let f a positive function such that $f>0$ for $y>a$ and $f(a)=0$ and let $I(x)$= $\int_{a}^{x}f(y)dy$ for $x $belongs to $ \left [ a,b \right ]$. Is that the function I(x) different to zero? Is that the function I(x) admits a minimum different to zero? – 2017-01-18