00:01
Hi there, so for this problem we are given the following function, that function f of x that is equal to x divided by x squared plus 16 and then the value of x in this case is between the interval of 0 and 7.
00:18
So the question is to find the absolute minimum value and the absolute maximum value for this given interval.
00:24
Now to find the maximum or the minimum of a function we need to first derivate that function with respect to the variable x in this case.
00:31
We can treat this as a product between two functions x and then the function in the denominator that is x squared plus 16.
00:39
First we derivate x so we will obtain just simply 1 divided by x squared plus 16.
00:47
Then this minus, then we leave x as a constant, then this divided by x squared plus 16 and that to the square, then we set this equal to 0 because we want to minimize this function.
00:59
Now we start solving in here for x.
01:02
So we can move this to the other side and multiply everything by x squared plus 16 to the square.
01:08
So we obtain that x is equal to, when we do that we obtain that this is x squared plus 16 in here.
01:19
Now we can move this x to the other side so we obtain 0 is equal to x squared minus x plus 16.
01:26
As you can recognize in here this is a quadratic equation, okay? so we can use the quadratic formula and we will obtain two possible values for this.
01:37
Oh sorry i have made a mistake in here, let me just fix that in there.
01:41
When we do the derivative of this i forgot the internal derivative which is the derivative of x squared.
01:47
So we need to multiply this by 2 times x.
01:50
Now okay and that's what we needed to fix in there, okay? so then we have something similar but in this case we will have 2 times x squared is equal to x squared plus 16.
02:03
Now we can pass this to this side so we will have 0 is equal to minus x squared plus 16...