Given-a-function-f-satisfy-f-x-3f-x-If-1-2-f-x-dx-2-then-2-1-f-x-dx-

Question Number 128750 by bemath last updated on 10/Jan/21

Given a function f satisfy f (- x) = 3 f (x) .

If \int_{- 1}^{2} f (x) dx = 2 then \int_{- 2}^{1} f (x) dx = ?

Commented by mr W last updated on 10/Jan/21

i f f (- x) = 3 f (x) i s v a l i d f o r a n y x \in R,

t h e n q u e s t i o n i s w r o n g, s i n c e f (x) = 0

a n d \int_{- 1}^{2} f (x) d x = 0 \neq 2 .

i f f (- x) = 3 f (x) i s v a l i d f o r x \in R^{+},

t h e n y o u {c a n}^{'} t d e t e r m i n e \int_{- 2}^{1} f (x) d x

f r o m t h e s i n g l e c o n d i t i o n

\int_{- 1}^{2} f (x) d x = 2 .

s o p l e a s e r e c h e c k t h e q u e s t i o n!

Answered by bramlexs22 last updated on 10/Jan/21

replace x by - x then we find

\int_{- 2}^{1} f (x) dx = \int_{2}^{- 1} f (- x) d (- x)

= \int_{- 1}^{2} f (- x) dx = \int_{- 1}^{2} 3 f (x) dx

= 3 \times 2 = 6 .

Commented by mr W last updated on 10/Jan/21

i f f (- x) = 3 f (x) f o r x \in R, t h e n

f (x) = 3 f (- x), a n d t h e n

f (x) = 3 f (- x) = 9 f (x), a n d t h e n

f (x) = 0, a n d t h e n

\int_{- 2}^{1} f (x) dx = 0, \int_{- 1}^{2} f (x) dx = 0 \neq 2 .

Commented by bramlexs22 last updated on 10/Jan/21

hahaha \dots . it follows that question wrong