I have to find where y is increasing/decreasing, where it's concave up/down and any inflection points, if there are horizontal asymptotes, and then sketch it. Here's the problem:
y = x^k/1+x^k
k is a positive integer greater than one. Here's what I get for the first derivative, after that I'm stuck:
My guess is that the function is increasing for x>o and decreasing for x<0 but stuck after that.
I don't really understand. I'm differentiating with respect to x. So I wanna find the derivative for x^k to find where the function increases/decreases and then find the derivative of that to find where it's concave up/down.
I thought the derivative of x^k would be kx^k-1. I dunno what the second derivative would be.
If x>0, there is no problem.
If x<0, it will be positive if k-1 is positive. Otherwise, it will be negative.
For the second derivative, it's the same thing. You have , which has as derivative...
I'm guessing you mean this:
So, using the quotient rule to find the derivative:
Derivate the terms on the right:
Simplify a little bit:
Convert the form to make your life easier for the next derivative:
Now take the next derivative using the product rule:
Derivate the first term using the chain rule:
And then the second term:
Simplify a bit:
That should get you started.