Thread: Concavity & Points of Inflection

1. Concavity & Points of Inflection

Let $f(x)$ be defined for all real numbers except at $x=0$ such that whose derivatives are given by
$f'=\frac{-e^\frac{1}{x}}{x^2}$ and $f"=\frac{e^\frac{1}{x}(2x+1)}{x^4}$
a) Find the interval(s) of concave upward/concave downward of $f(x)$
b) At what value(s) of x does the function $f(x)$ have point(s) of inflection?

2. Originally Posted by drahcirnaw
Let $f(x)$ be defined for all real numbers except at $x=0$ such that whose derivatives are given by
$f'=\frac{-e^\frac{1}{x}}{x^2}$ and $f"=\frac{e^\frac{1}{x}(2x+1)}{x^4}$
a) Find the interval(s) of concave upward/concave downward of $f(x)$
b) At what value(s) of x does the function $f(x)$ have point(s) of inflection?
a) Find the interval(s) of concave upward/concave downward of $f(x)$

$f(x)$ is concave upward when $f"(x)>0$

$f(x)$ is concave downward when $f"(x)<0$

For $x\neq0$, $e^\frac{1}{x}>0$ and $x^4>0$ so

$f"(x)>0$ when $2x+1>0$ and

$f"(x)<0$ when $2x+1<0$

b) At what value(s) of x does the function $f(x)$ have point(s) of inflection?

$f(x)$ has a point of inflection where it changes concavity