00:01
For this problem, we are asked to determine the intervals of constant concavity of the function f of x equals xe to the power of x, as well as to locate any inflection points.
00:10
Now, our first step here will be to take the first derivative of our function.
00:14
To do this, we'll have to apply the product rule.
00:17
We'll find that this would be then e to the power of x plus xe to the power of x.
00:21
Now taking the second derivative, we find that this would then be e to the power of x plus e to the power of x plus xe to the power of x.
00:30
Or we could write this as e to the power of x times 2 plus x.
00:36
Now we want to solve for when this will be equal to 0.
00:40
We can see that we'd either need e to power of x to equal 0, which is never true, or we need 2 plus x to be equal to 0...