00:01
Consider the function f of x equal negative 2 x to the 3 half plus 9x plus 15.
00:14
We want to find the critical values of this function.
00:19
That is the critical numbers.
00:21
The values of x for which the derivative is 0 or does not exist.
00:28
Ok, so let's find the derivatives.
00:31
First of all, let's say this function is defined for positive values of x or non -negative because 0 is also defined.
00:40
But x can be non -negative, cannot be negative because x to the 3 half is the same as the square root of x cubed.
00:53
And x cubed being negative for the negative values of x, the square root won't be a real number in that case.
01:02
So, we can say that the domain of f is the closed interval at 0 and open to the right at plus infinity.
01:17
So, that is for the real numbers such that x is greater than or equal to 0.
01:26
At 0 is well defined, is equal to 15.
01:30
But it's not defined for the negative numbers.
01:33
So, all critical values of the function must be in that interval.
01:39
Indeed, on the open interval 0 plus infinity.
01:44
Another thing is very important to notice that this is not a polynomial function because we have this exponent 3 half here.
01:54
So, it's not a polynomial function.
01:59
Good, so let's calculate the derivative which is essential then to find the critical values of f.
02:06
That is negative 2 is a constant times derivative of x to the 3 half is 3 half times x to the 3 half minus 1.
02:19
And that's all because the base x is only the variable x, not a function of x.
02:25
So, we stop there for that term negative 2x to the 3 half...