00:01
In this problem, the first question is to graph a polynomial that has two local maxima and two local minima.
00:16
Now, consider the function f of x.
00:19
The extreme values of the function occurs at the points where the derivative f dash of x of the function is equal to 0.
00:27
And these x values corresponds to the critical points of the function.
00:31
And the extreme values occur at the critical points of the function.
00:36
And the extreme values can be maximum or minimum, depending upon whether the second derivative at the critical point is less than 0 or greater than 0.
00:49
If the second derivative at the critical point is less than 0, then at this point the function will have a local maximum.
00:56
And if the second derivative is greater than 0 at the critical point, then at that point the function, have a local minimum.
01:04
Now we need to graph a polynomial with two local maximum and two local minimum that means it must be a polynomial with four critical points and these critical points must be distinct.
01:20
That means it must be a polynomial such that its derivative will have four distinct real roots.
01:30
That means it must be a polynomial of degree 5 so that its derivative f dash of x will be a polynomial with degree 4 and a 4th degree polynomial can have a maximum of 4 real roots thus graph a 5th degree polynomial that has 4 distinct real roots for its derivative, such a polynomial will have two local maximum and two local minimum.
02:10
This is the graph of a 50 degree polynomial name as f of x and see that it has critical points at negative 2, negative 1, 1 and 2 and the points x is equal to negative 2 and x is equal to 1 are points of local maximum for this function and the point.
02:34
Means x is equal to negative 1 and x is equal to 2 are points of local minima for this function.
02:40
The local extreme values are also mapped in this graph...