00:01
In this question, the graph of the derivative f -tash x of a function f of x is given as this figure.
00:14
Using this graph, we need to obtain the, for the first subpart, we need to obtain the values of x for which the function f of x is increasing.
00:28
Now, the function f of x increases for the values for which the derivative is greater than 0.
00:40
That is, all the values of f of x which are above the y -axis, or in the positive direction of y -axis, will have f -f x as increasing.
00:52
That means from 2 to 4, from x is equal to 2, x is equal to 4, values, values, of f -dash is increasing.
01:05
F -dash -x is greater than 0.
01:10
That means in the interval 2 to 4, f of x would be increasing.
01:15
Also, in the interval 6 to 9.
01:26
F -dash of x is greater than 0, that is positive.
01:30
Therefore, the function will also increase in the interval 6 to 9.
01:36
Therefore, this is the solution for the first sub -part.
01:41
For the second subpart, we need to obtain the values of x for which the function is decrease.
01:52
Now, f of x would decrease for the value for which f -dash of x is less than 0.
02:02
That is negative.
02:04
Now, from the graph we can see that f -pash x is negative in the interval 0 to 2 and in the interval 4 to 6.
02:15
Therefore, f -tash -x is less than equal to 0 from 0 to 2 and from 4 to 6.
02:29
Therefore, the function will be decreasing in these two intervals.
02:35
As this is the solution for the second subpart.
02:40
For the third subpart, we need to obtain the values of x at which the function will have its local maximum.
02:51
Now, the function has local maxima where the derivative of the function goes from positive to negative.
03:03
That means the change in function goes from increasing to decreasing.
03:09
Now, from the graph we can see that at this point four, the value of f -dash -x goes from positive to negative...