Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
Suppose that we know that 1 is greater than or equal to g(x)dx with bounds 0 to 4 and is equal or greater to 8. Determine the best possible upper and lower bound for x g(x^2)dx with bounds 0 to 2 that can be deduced from this information.
This is just a glorified integration by substitution problem.
Consider
Let
Changing the limits we find:
when ,
when ,
So our integral becomes:
I leave the easy part to you
EDIT: Well thanks for making me type all this, Moo