×

# Is f(x)=(x-1/x) concave or convex at x=-1?

Feb 10, 2017

Since $f ' ' \left(- 1\right) = 2 > 0$, the function is convex (commonly called concave up) at $x = - 1$.

#### Explanation:

The convexity and concavity of a function can be determined through its second derivative. At $x = a$, a function $f$ is:

• convex (commonly known as concave up) if $f ' ' \left(a\right) > 0$
• concave (commonly known as concave down) if $f ' ' \left(a\right) < 0$

Find the function's second derivative by rewriting with a negative power then using the power rule:

$f \left(x\right) = x - \frac{1}{x}$

$f \left(x\right) = x - {x}^{-} 1$

$f ' \left(x\right) = 1 - \left(- 1 {x}^{-} 2\right)$

$f ' \left(x\right) = 1 + {x}^{-} 2$

$f ' ' \left(x\right) = - 2 {x}^{-} 3$

$f ' ' \left(x\right) = - \frac{2}{x} ^ 3$

The value of the second derivative at $x = - 1$ is:

$f ' ' \left(- 1\right) = - \frac{2}{- 1} ^ 3 = - \frac{2}{- 1} = 2$

Since $f ' ' \left(- 1\right) = 2 > 0$, the function is convex (commonly called concave up) at $x = - 1$.