if-f-0-1-f-1-2-and-0-1-f-x-dx-3-than-0-1-x-f-x-dx-a-1-b-1-c-2-d-2-

Question Number 95800 by rb222 last updated on 27/May/20

i f f (0) = 1

f (1) = 2 a n d \int_{0}^{1} f (x) d x = 3

t h a n \int_{0}^{1} x f (x) d x = ?

a . 1

b . - 1

c . 2

d . - 2

Commented by mr W last updated on 27/May/20

n o u n i q u e s o l u t i o n!

Commented by rb222 last updated on 27/May/20

t h a n k s s i r

Answered by bobhans last updated on 27/May/20

f (x) = {px}^{2} + qx + 1 \Rightarrow f (1) = p + q = 1

p = 1 - q \Rightarrow \underset{0}{\int^{1}} ((1 - q) x^{2} + qx + 1) dx = 3

\frac{1 - q}{3} + \frac{q}{2} + 1 = 3; \frac{2 + q}{6} = 2 \Rightarrow q = 10

p = - 9 . \Rightarrow \underset{0}{\int^{1}} xf (x) dx = \underset{0}{\int^{1}} x (- {9 x}^{2} + 10 x + 1) dx

= \underset{0}{\int^{1}} (- {9 x}^{3} + {10 x}^{2} + x) dx = \frac{- 9}{4} + \frac{10}{3} + \frac{1}{2}

= \frac{- 27 + 40 + 6}{12} = \frac{19}{12}

Commented by mr W last updated on 27/May/20

f (x) c a n b e a n y t y p e o f f u n c t i o n s .

Commented by bobhans last updated on 28/May/20

oo yes sir

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