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Math Help - Concavity & Points of Inflection

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    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?
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    Quote Originally Posted by drahcirnaw View Post
    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
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