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Thread: Concavity & Points of Inflection

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    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?
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    Quote Originally Posted by drahcirnaw View Post
    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
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