proof of monotonicity criterion
Let us start from the implications “”.
and being we conclude that .
This proves the first claim. The other three cases can be achieved with minor modifications: replace all “” respectively with , and .
Let us now prove the implication “” for the first and second statement.
Given consider the ratio
If is increasing the numerator of this ratio is when and is when . Anyway the ratio is since the denominator has the same sign of the numerator. Since we know by hypothesis that the function is differentiable in we can pass to the limit to conclude that
If is decreasing the ratio considered turns out to be hence the conclusion .
Notice that if we suppose that is strictly increasing we obtain the this ratio is , but passing to the limit as we cannot conclude that but only (again) .
|Title||proof of monotonicity criterion|
|Date of creation||2013-03-22 13:45:14|
|Last modified on||2013-03-22 13:45:14|
|Last modified by||paolini (1187)|