if-function-f-satisfy-form-f-x-0-1-f-x-dx-x-2-0-2-f-x-dx-x-0-3-f-x-dx-1-then-the-value-of-f-4-is-

Question Number 2705 by Syaka last updated on 25/Nov/15

i f f u n c t i o n f s a t i s f y f o r m :

f (x) = (\underset{0}{\int^{1}} f (x) d x) x^{2} + (\underset{0}{\int^{2}} f (x) d x) x + (\underset{0}{\int^{3}} f (x) d x) + 1

t h e n t h e v a l u e o f f (4) i s \dots . . ?

Answered by prakash jain last updated on 25/Nov/15

f (x) = {a x}^{2} + b x + c \dots . (1)

\int f (x) = a \frac{x^{3}}{3} + b \frac{x^{2}}{2} + c x + C

\int_{0}^{1} f (x) = \frac{a}{3} + \frac{b}{2} + c

\int_{0}^{2} f (x) = \frac{8 a}{3} + 2 b + 2 c

\int_{0}^{3} f (x) = 9 a + \frac{9}{2} b + 3 c

f (x) =

(\frac{a}{3} + \frac{b}{2} + c) x^{2} + (\frac{8 a}{3} + 2 b + 2 c) x + (9 a + \frac{9}{2} b + 3 c) + 1 . . (2)

E quating coefficients in (1) and (2)

(x^{2} c o e f f i i e n t)

\frac{a}{3} + \frac{b}{2} + c = a \Rightarrow \frac{2 a}{3} = \frac{b}{2} + c \Rightarrow a = \frac{3}{4} b + \frac{3}{2} c

(x c o e f f i c i e n t)

\frac{8 a}{3} + 2 b + 2 c = b \Rightarrow 2 b + 4 c + 2 b + 2 c = b \Rightarrow

5 b = - 6 c \Rightarrow b = \frac{- 6 c}{5}

a = \frac{3 b}{4} + \frac{3 c}{2} \Rightarrow a = - \frac{3}{4} \times \frac{- 6 c}{5} + \frac{3 c}{2} = \frac{12 c}{20} = \frac{3 c}{5}

(c o n s t a n t t e r m)

(9 a + \frac{9}{2} b + 3 c) + 1 = c

To be solved for a, b and c

\frac{27 c}{5} - \frac{9}{2} \times \frac{6 c}{5} + 2 c = - 1

c = \frac{- 1}{2}, a = \frac{- 3}{10}, b = \frac{3}{5}

f (x) = - \frac{3}{10} x^{2} + \frac{3}{5} x - \frac{1}{2}

f (4) = \frac{- 3}{10} \times 16 + \frac{3}{5} \times 4 - \frac{1}{2} = \frac{- 48 + 24 - 5}{10} = \frac{- 29}{10}

Commented by prakash jain last updated on 25/Nov/15

You can also start with

f (x) = {a x}^{2} + b x + (c + 1)

As long as you treat (c + 1) as constant term .

Commented by Syaka last updated on 25/Nov/15

I s f (x) a r e f (x) = {a x}^{2} + b x + c + 1, {i s n}^{'} t ?

Commented by prakash jain last updated on 25/Nov/15

For a polynomial c a n d 1 b o t h a r e c o n s t a n t s

s o w e u s e {a x}^{2} + b x + c a n d w h i l e c o m p a r i n g

c o m p a r e w i t h \int_{0}^{3} f (x) d x + 1 .