If f and g are continuous functions and the integral from 0 to 1 of f(x) dx < the integral from...?

Question

...0 to 1 of g(x) dx, then f(x) <= g(x) for all x in (0,1).

Is the statement false?

Thanks!

Answer 1

If f(x) < g(x) for every x in an interval [a, b], then Int (a to b ) f(x) dx <= Int (a to b) g(x) dx . But, as seen in Mathematician example, Int (a to b ) f(x) dx < Int (a to b) g(x) dx does NOT imply f(x) < g(x) (nor f(x) <- g(x) in (a, b).

Answer 2

False - but dont know a nice proof.

If you think of the integral as the area under each function, then just because the area of g(x) is larger than f(x) does not mean that the function itself necessarily is always larger.

Answer 3

Flase. Let f(x)=1/4 (constant) and g(x)=x.