Question-169631

Question Number 169631 by MaxiMaths last updated on 05/May/22

Commented by MaxiMaths last updated on 05/May/22

any way to find the function f ? ? ?

Commented by mr W last updated on 05/May/22

y o u m e a n t - 4 a t x = - 1 ?

Commented by mr W last updated on 05/May/22

i n f i n i t e p o s s i b i l i t i e s .

e x a m p l e f (x) = - \frac{14}{9} (\frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 x) - \frac{59}{27}

Commented by mr W last updated on 05/May/22

Answered by mr W last updated on 05/May/22

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

f (x) = \int g (x) (x + 1) (x - 2) + C

f (- 1) = - 4, f (2) = 3

\Rightarrow t h e r e a r e i n f i n i t e p o s s i b i l t i e s f o r

g (x) a n d C .

t h e m o s t s i m p l e c a s e i s

g (x) = a = c o n s t a n t .

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

f (x) = a (\frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 x) + b

f (- 1) = a (- \frac{1}{3} - \frac{1}{2} + 2) + b = - 4

\Rightarrow \frac{7}{6} a + b = - 4

f (2) = a (\frac{8}{3} - \frac{4}{2} - 4) + b = 3

\Rightarrow - \frac{10}{3} a + b = 3

\Rightarrow a = - \frac{14}{9}

\Rightarrow b = - \frac{59}{27}

\Rightarrow f (x) = - \frac{14}{9} (\frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 x) - \frac{59}{27}

Commented by MaxiMaths last updated on 06/May/22

waoh! good!