Category: Studying Changes of Function
Condition of Constancy of the Function
Fact. Suppose function is defined and continuous on interval and has finite derivative . Function is constant if and only if for all in .
Condition of Monotony of the Function
Now let's formulate conditions when function will be monotonic, i.e. increasing, decreasing, non-decreasing, non-increasing.
Fact 1. Suppose function is defined and continuous on interval and has finite derivative on it. Then function is non-decreasing (non-increasing) on if and only if for all in .
Proving Inequalities Using Monotony of the Function
Monotony of the function can be used to prove some not obvious inequalities.
Example 1. Prove that when .
Consider function defined on interval .
Maxima and Minima. Necessary Conditions
If function is defined and continuous on interval is not monotonic, then exist such subintervals of interval that largest and smallest value function takes at inner point, i.e. between and .
First Derivative Test
We know that if point is extremum then it is stationary point. So, we need to find all stationary points and test them. But how?
For this we use following test.
First Derivative Test. Suppose that is a stationary point of a continuous function .
Second Derivative Test
Instead of using First Derivative Test we can use another test.
Second Derivative Test.
- if and then there is maximum at point .
- if and then there is minimum at point .
- if and then we can't say anything about point .
This test is used not so often as first derivative test because of two reasons:
Higher-Order Derivative Test
As we know Second derivative test is inconlusive when for critocal point .
In this case we need to use another test.
Higher-Order Derivative Test. Suppose that is the smallest number for which . If is and odd number, then there is no maximum or minimum at . If is even number then is maximum if , and is minimum if .
Closed Interval Method
In fact when we used derivative to find extrema, this extrema mainly were local.
Now we turn our attention to finding global extrema on closed interval .
Extreme Value Theorem. If is continuous on a closed interval , then attains a global maximum value and a global minimum value at some numbers and from interval .