For what values of x is f(x)= x^2-x + 1/x-1/x^2 f(x)=x2x+1x1x2 concave or convex?

1 Answer
Nov 19, 2017

x=-1, 3, (3+-sqrt(21))/2

Explanation:

Find the second derivative of f(x).
f'(x)=2x-1-1/x^2+2/x^3
f''(x)=2+2/x^3-6/x^4

Find the critical points of f''(x).
0=2+2/x^3-6/x^4
0=2(1+1/x^3-3/x^4)
-1=1/x^3-3/x^4
-1=x^3-x^4/3
-1=x(x^2-x^3/3)

x=-1

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

x=3

x^2-3x=3
0=x^2-3x-3

Use the quadratic formula.

x=(-b+-sqrt(b^2-4ac))/(2a)=(3+-sqrt(21))/2

x=-1, 3, (3+-sqrt(21))/2