JEE Advance - Mathematics (1989 - No. 29)
If $$f$$ and $$g$$ are continuous function on $$\left[ {0,a} \right]$$ satisfying
$$f\left( x \right) = f\left( {a - x} \right)$$ and $$g\left( x \right) + g\left( {a - x} \right) = 2,$$
then show that $$\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $$
$$f\left( x \right) = f\left( {a - x} \right)$$ and $$g\left( x \right) + g\left( {a - x} \right) = 2,$$
then show that $$\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $$
This statement is false.
$$\int_0^a f(x)g(x) dx = \int_0^a f(x) dx$$
$$\int_0^a f(x)g(x) dx = 2\int_0^a f(x) dx$$
$$\int_0^a f(x)g(x) dx = \frac{1}{2}\int_0^a f(x) dx$$
$$\int_0^a f(x)g(x) dx = 0$$
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