00:01
For this problem, we are to find the point of inflection of f of x, which is equal to arc tangent of x squared.
00:09
And then we describe the concavity of this function.
00:13
First, we want to get the derivatives of f until the second derivative.
00:17
Now, f prime of x by the chain rule is equal to 1 over 1 plus the square of x squared, or that's 1 over 1 plus x raise to the 4th.
00:30
Power so f double prime of x is equal to negative 4x cubed over 1 plus x -rays to the 4th power squared.
00:42
And then from here we want to set the second derivative equal to 0 and we solve for x to get the possible inflection points.
00:54
So we have negative 4x cubed over the square of 1 plus x -rays to the 4th power squared equal 0.
01:01
That will imply that x is equal to zero and looking at the form of the second derivative we know that the denominator can never be zero so this is the only point we can have to be the possible inflection point now before we can determine if this will form an inflection for the graph of the function we first have to see if the concavity or concavities of f of x changes at this point.
01:40
So what we're going to do is we partition the domain of f of x using x equals 0.
01:48
Now the domain for arc tangent of x squared is all real numbers.
01:54
So we want to partition the real number line using x equals 0.
01:57
So these are the intervals that we have negative infinity to 0 and 0 to infinity...