Determining concavity of f(x) = x^2 + 2x/(x-2)^2

[Bob Ho]

Hi, sorry to disturb you,
But with the equation f(x)= x^2+2x/(x-2)^2

I need to find the intervals at which f(x) is concave up, and down.

I found f '(x)= (2x+2)(x-2)^2 -2(x-2)(x^2+x)/((x-2)^4)

From there I equated it to 0, and found the critical points to be x=2,-2/3.

However, I have just noticed there is a point of inflection at x=-2. How did I miss this point in my calculation for critical points, any help would be much obliged.

Thanks

 

[Dick]

A point of inflection is not necessarily a critical point. To know about concavity and points of inflection you need to compute the second derivative.

 

[Defennder]

To find points of inflection, find the values of x for which f''(x) is zero.

 

[HallsofIvy]

More specifically, where the second derivative changes sign. A point where the second derivative is 0 may not be a point of inflection.