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Been stuck on this one for a while now. Any help would be appreciated:

Let $f:[a,b]\rightarrow\mathbb R$ be an integrable function. Prove that there exist a $c\in[a,b]$ such that:

$$\int_a^c f(x)dx=\int_c^b f(x)dx$$

Orokusaki
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  • Can you use the fact that the indefinite integral, $\int_a^x f(t)dt$, is a continuous function? If so then you can use the "intermediate value property". – user247327 Apr 15 '20 at 11:55
  • Yes, it'ts implied from the fundamental theorem of calculus. I see now how I can use the intermediate value property, thanks – Orokusaki Apr 15 '20 at 14:28

1 Answers1

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Let $F$ be defined by $F(x):=\int_a^x f(t)dt -\int_x^b f(t) dt$ for $a \le x \le b.$

Then we have:

$F$ is continuous, $F(b)=\int_a^b f(t)dt$ and $F(a)=-\int_a^b f(t)dt=-F(b).$

Hence there is $c \in [a,b]$ such that $F(c)=0$.

Fred
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