Given-function-f-R-R-x-2-f-x-f-1-x-2x-x-4-f-2019-

Question Number 79532 by naka3546 last updated on 26/Jan/20

G i v e n f u n c t i o n f : R \Rightarrow R

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

f (2019) = ?

Commented by john santu last updated on 26/Jan/20

(1) x = 2019

{(2019)}^{2} f (2019) + f (- 2018) = 2 \times (2019) - {(2019)}^{4}

(2) x = - 2018

{(- 2018)}^{2} f (- 2018) + f (2019) = 2 \times (- 2018) - {(- 2018)}^{4}

n o w y o u g e t t h e r e s u l t

Commented by mathmax by abdo last updated on 26/Jan/20

x^{2} f (x) + f (1 - x) = 2 x - x^{4} l e t c h a n g e x b y 1 - x \Rightarrow

(1 - x) f (1 - x) + f (x) = 2 (1 - x) - {(1 - x)}^{4} s o w e g e t t h e s y s t e m e

{\begin{cases} x^{2} f (x) + f (1 - x) = 2 x - x^{4} \\ f (x) + {(1 - x)}^{2} f (1 - x) = 2 (1 - x) - {(1 - x)}^{4} \end{cases}

Δ_{s} = | \begin{matrix} x^{2} 1 \\ 1 {(1 - x)}^{2} \end{matrix} | = x^{2} {(1 - x)}^{2} - 1 \Rightarrow f (x) = \frac{| \begin{matrix} 2 x - x^{4} 1 \\ 2 (1 - x) - {(1 - x)}^{4} {(1 - x)}^{2} \end{matrix} |}{x^{2} - x^{3} - 1}

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

f (2019) = \frac{(2 \times 2019 - {(2019)}^{4}) {(2018)}^{2} + 2 \times (2018) + {(2018)}^{4}}{{(2019)}^{2} \times 2018^{2} - 1}

Answered by mr W last updated on 26/Jan/20

x^{2} f (x) + f (1 - x) = 2 x - x^{4} \dots (i)

l e t t = 1 - x \Rightarrow x = 1 - t

{(1 - t)}^{2} f (1 - t) + f (t) = 2 (1 - t) - {(1 - t)}^{4}

o r

{(1 - x)}^{2} f (1 - x) + f (x) = 2 (1 - x) - {(1 - x)}^{4} \dots (i i)

(i) \times {(1 - x)}^{2} - (i i) :

[x^{2} {(1 - x)}^{2} - 1] f (x) = 2 x {(1 - x)}^{2} - 2 (1 - x) + {(1 - x)}^{4} - x^{4} {(1 - x)}^{2}

[x^{2} {(1 - x)}^{2} - 1] f (x) = (x - 1) {2 + 2 (x - 1) x + {(x - 1)}^{3} - (x - 1) x^{4}}

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

f (2019) = \frac{2018 {2 + 2 \times 2018 \times 2019 + 2018^{3} - 2018 \times 2019^{4}}}{2018^{2} \times 2019^{2} - 1}

f (2019) = - 4076360

Facebook Tweet Pin