Is f(x) =(x+2)^2/(x-1) concave or convex at x=-1?

1 Answer
Nov 27, 2017

concave

Explanation:

Let's calculate the second derivative and then find the sign of it when x=-1:

The first derivative is:

f'(x)=(2(x+2)(x-1)-(x+2)^2)/(x-1)^2

=((x+2)(2x-2-x-2))/(x-1)^2=((x+2)(x-4))/(x-1)^2

=(x^2-2x-8)/(x-1)^2

The second derivative is:

f''(x)=((2x-2)(x-1)^2-(x^2-2x-8)*2(x-1))/(x-1)^4

=(2(x-1)^3-2(x-1)(x^2-2x-8))/(x-1)^4

=(2cancel((x-1))(cancelx^2cancel(-2x)+1cancel(-x^2)+cancel(2x)+8))/(x-1)^(cancel4^3)

=18/(x-1)^3

that's less than 0 at x=-1:

f''(-1)=18/(-1-1)^3=18/-8=-9/4<0

Then the given function is concave at x=-1

graph{(x+2)^2/(x-1) [-10, 10, -20, 20]}