# Thread: Concavity & Points of Inflection

1. ## Concavity & Points of Inflection

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

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

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

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

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

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

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

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

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