For what values of x is f(x)=(-2x^2)/(x-1)f(x)=2x2x1 concave or convex?

1 Answer

The function will be concave in (1, \infty)(1,)

The function will be convex in (- \infty, 1)(,1)

Explanation:

Given function:

f(x)=\frac{-2x^2}{x-1}f(x)=2x2x1

f'(x)=\frac{(x-1)(-4x)-(-2x^2)(1)}{(x-1)^2}

=\frac{-2x^2+4x}{(x-1)^2}

f''(x)=\frac{(x-1)^2(-4x+4)-(-2x^2+4x)(2(x-1))}{(x-1)^4}

f''(x)=-4/(x-1)^3

The function will be concave iff f''(x)<0

\therefore -4/(x-1)^3< 0

1/(x-1)^3> 0

x\in (1, \infty)

The function will be concave in (1, \infty)

The function will be convex in (- \infty, 1)