# Thread: Does this necessarily follow?

1. ## Does this necessarily follow?

Assume that f and g are continuous on [a,b] and

$\int_a^b f(x) dx > \int_a^b g(x) dx$

Does it necessarily follow that $f(x)>g(x)$ for all $x\in[a,b]$?

2. No
Take for instance f(x) = 1/2 and g(x) = x over [-1,1]

3. No, put $f(x)=\frac1{1+x^2}$ and $g(x)=x$ for $0\le x\le1.$

But $\frac1{1+x^2}\not>x.$ (At least not always.)

(Anyway, the converse is true.)

4. $If\ f(x)>g(x)\ for\ all\ x\ over\ an\ interval,$

$then\ \int{f(x)dx}>\int{g(x)dx}\ certainly,\ over\ the\ interval.$

The converse is not true however.
Similar to the graph given by running-gag,

$f(x)=4x\ from\ x=0\ to\ x=1$

when integrated, gives 2.

$g(x)=x+1$

when integrated over the same interval gives $\frac{3}{2}$

$f(x)

but

$f(x)>g(x)\ from\ x=\frac{1}{3}\ to\ x=1.$