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Let $f$ and $g$ be continuous functions on $[a,b]$ and $g>0$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, \mathrm{d}x = f(c)\int_a^b g(x) \, \mathrm{d}x.$$

Klaus
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yamita kumari
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    This is a standard result called the "mean value theorem for integrals". – Ian Feb 21 '19 at 15:15
  • Welcome! Please try and format your posts properly in the future https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – siegehalver Feb 21 '19 at 15:22

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Hint

There exist $m,M\in [a,b]$ such that $f(m)\le f(x)\le f(M), \forall x\in [a,b].$ Since $g>0$ we have $$f(m)g(x)\le f(x)g(x)\le f(M)g(x), \forall x\in [a,b].$$

mfl
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