If $f\colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ Then there is a $ c \in [a,b] $ which have this equation for

Question

I’m reading a textbook which poses this theorem but I don’t get where it came from .

If $f\colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ Then there is a $c\in [ a,b] $ such that :

$$ \int_a^c f(x) \ \mathrm{d} x = \int_c^b f(x) \ \mathrm{d} x $$

Thanks in advance.

Answer 1

Hint: Apply the intermediate value theorem to the map$$c\mapsto\int_a^cf(x)\,\mathrm dx-\int_c^bf(x)\,\mathrm dx.$$

Answer 2

Hint:

Application of the intermediate value theorem to

$$F(x) = \int_a^x f - \int_x^b f $$

Answer 3

The theorem is not true , if one requiers $c\in ]a,b[$ !

Example: $a=0, b=2$ and $f(x)=1-x$.

Then we have $\int_a^c f(x) \ \mathrm{d} x = \int_c^b f(x) \ \mathrm{d} x \iff c=0$ or $c=2$.

Reason: $\int_0^2 (1-x) \ \mathrm{d} x = 0.$

If we have $\int_a^b f(x) \ \mathrm{d} x \ne 0$, then the theorem is true and can be proved as Jose Carlos suggedted.