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Thread: Find the intervals on which f(x) = ln(x + 1) is concave up

  1. #1
    VkL
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    Find the intervals on which f(x) = ln(x + 1) is concave up

    Find the intervals on which f(x) = ln(x + 1) is concave up

    I got critical points to be -1, and 1

    I checked by the first derivative that from - infinity to 0 the slope is negative and from 0 to positive infinity the slope is positive, i get confused about writing the intervals.
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    Quote Originally Posted by VkL View Post
    Find the intervals on which f(x) = ln(x + 1) is concave up

    I got critical points to be -1, and 1

    I checked by the first derivative that from - infinity to 0 the slope is negative and from 0 to positive infinity the slope is positive, i get confused about writing the intervals.
    We need to find the second derivative. First $\displaystyle f'(x) = \tfrac{(x^2+1)'}{x^2+1} = \tfrac{2x}{x^2+1}$ by the chain rule.
    Second, $\displaystyle f''(x) = \tfrac{(2x)'(x^2+1) - (2x)(x^2+1)'}{(x^2+1)^2} = \tfrac{2(1-x^2)}{(x^2+1)^2} $. It concaves up when $\displaystyle f''(x) > 0$ so $\displaystyle \tfrac{2(1-x^2)}{(x^2+1)^2} > 0 $. Multiply this inequality both sides by $\displaystyle \tfrac{(x^2+1)^2}{2}$ to get: $\displaystyle \tfrac{(x^2+1)^2}{2} \cdot \tfrac{2(1-x^2)}{(x^2+1)^2} > 0 \cdot \tfrac{(x^2+1)^2}{2}$*. Thus, $\displaystyle 1-x^2 > 0 \implies x^2 < 1 \implies -1 < x < 1$.


    *)Remember that $\displaystyle \tfrac{(x^2+1)^2}{2} > 0$ so the inequality does not flip.
    ----

    There is another way to do this problem, a more straightforward way that you probably learned at college but a little longer. First, solve $\displaystyle f''(x) = 0$ and that happens when $\displaystyle \tfrac{2(1-x^2)}{(x^2+1)^2} = 0 \implies 1-x^2 = 0 \implies x = \pm 1$. Now put the points $\displaystyle x=-1, x=1$ on the number line. Take any point to the left of $\displaystyle -1$, say $\displaystyle -2$ and compute $\displaystyle f''(-2) < 0$ ---> thus $\displaystyle f$ concaves down for $\displaystyle -\infty < x < -1$. Take any point between $\displaystyle -1$ and $\displaystyle 1$, say $\displaystyle 0$ and compute $\displaystyle f''(0) > 0 $ ---> thus $\displaystyle f$ concave up for $\displaystyle -1 < x < 1$. Take any point to the right of $\displaystyle 1$, say $\displaystyle 2$ and compute $\displaystyle f''(2) < 0$ ---> thus $\displaystyle f$ concaves down for $\displaystyle 1<x<\infty$.
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