If-f-x-x-3-3x-2-2-x-1-4-Then-1-4-3-4-f-f-x-dx-

Question Number 42709 by rahul 19 last updated on 01/Sep/18

If f (x) = x^{3} - \frac{3 x^{2}}{2} + x + \frac{1}{4} .

T hen \int_{\frac{1}{4}}^{\frac{3}{4}} f (f (x)) d x = ?

Commented by rahul 19 last updated on 01/Sep/18

Hint given : Replace x by 1 - x in

function and then arrange to get result .

Commented by rahul 19 last updated on 02/Sep/18

Thanks sir .

Commented by MJS last updated on 01/Sep/18

f (1 - x) = - x^{3} + \frac{3}{2} x^{2} - x + \frac{3}{4} = 1 - f (x)

f (f (1 - x)) = 1 - f (f (x))

\underset{\frac{1}{4}}{\int^{\frac{3}{4}}} f (f (1 - x)) d x = \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} 1 d x - \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} f (f (x)) d x

but \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} f (f (1 - x)) d x \underset{\frac{3}{4}}{\int^{\frac{1}{4}}} f (f (x)) d x = - \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} f (f (x)) d x

\Rightarrow 2 \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} f (f (x)) d x = \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} 1 d x \Rightarrow \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} f (f (x)) d x = \frac{1}{2} \underset{\frac{1}{4}}{\int^{\frac{3}{4}}} d x = \frac{1}{4}

Answered by Meritguide1234 last updated on 03/Sep/18

Commented by Meritguide1234 last updated on 03/Sep/18

i u s e w o r d t o t y p e m a t h s .

i l i v e i n t a i w a n .