Is $f (x) = (x - \frac{1}{x})$ $f(x)=(x-1/x)$ concave or convex at $x = - 1$ $x=-1$ ?

Question

3592 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-02-10T00:53:38

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

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

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

$f(x)=x-1/x$

$f(x)=x-x^-1$

$f'(x)=1-(-1x^-2)$

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

$f''(x)=-2x^-3$

$f''(x)=-2/x^3$

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

$f''(-1)=-2/(-1)^3=-2/(-1)=2$

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

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