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If f(x) and g(x) are continuous functions on the interval [0, 4] satisfying f(x) = f (4-x) and $g (x) + g (4 - x) = 3 and \int_{0}^{4} f (x) dx = 2 then \int_{0}^{4} f (x) g (x) dx =$

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answer is D.

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Detailed Solution

$I = \int_{0}^{4} f (x) g (x) dx = \int_{0}^{4} f (4 - x) g (4 - x) dx$

$= \int_{0}^{4} f (x) g (4 - x) dx = \int_{0}^{4} f (x) (3 - g (x)) dx = 3 \int_{0}^{4} f (x) d x - \int_{0}^{4} f (x) g (x) d x = 3 (2) - I \Rightarrow 2 I = 6 \Rightarrow I = 3$

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