Charlie Conner

2022-09-30

$f(x)$ is a differentiable function defined for x>2 satisfying

$${\int}_{3}^{ab+2}f\left(x\right)dx={\int}_{3}^{a+2}f\left(x\right)dx+{\int}_{3}^{b+2}f\left(x\right)dx$$

where $a,b\in (1,\mathrm{\infty})$) and find f(x)

$${\int}_{3}^{ab+2}f\left(x\right)dx={\int}_{3}^{a+2}f\left(x\right)dx+{\int}_{3}^{b+2}f\left(x\right)dx$$

where $a,b\in (1,\mathrm{\infty})$) and find f(x)

Toby Barron

Beginner2022-10-01Added 7 answers

f(x) is a differentiable function defined for x>2 satisfying

$${\int}_{3}^{ab+2}f\left(x\right)dx={\int}_{3}^{a+2}f\left(x\right)dx+{\int}_{3}^{b+2}f\left(x\right)dx$$

where $a,b\in (1,\mathrm{\infty})$ find $f(x)$

Take derivative to a

$$bf(ab+2)=f(a+2)$$

Let a=1

$$f(b+2)=\frac{f(3)}{b}$$

Let $x=b+2$

$$f(x)=\frac{f(3)}{x-2}$$

where $f(3)$ is some constant $c=f(3)$, then we have

$$f(x)=\frac{c}{x-2}$$

$${\int}_{3}^{ab+2}f\left(x\right)dx={\int}_{3}^{a+2}f\left(x\right)dx+{\int}_{3}^{b+2}f\left(x\right)dx$$

where $a,b\in (1,\mathrm{\infty})$ find $f(x)$

Take derivative to a

$$bf(ab+2)=f(a+2)$$

Let a=1

$$f(b+2)=\frac{f(3)}{b}$$

Let $x=b+2$

$$f(x)=\frac{f(3)}{x-2}$$

where $f(3)$ is some constant $c=f(3)$, then we have

$$f(x)=\frac{c}{x-2}$$

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