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Thread: Concave up and Concave down

  1. #1
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    Concave up and Concave down

    Hey

    I don't understand how to determine the domain of concave up versus concave down when the graph has two points of inflection.

    For example

    x^4 + 8x^3 + 18x^2 + 8

    extreme points: -3, 0
    (graph is high at -3 and low at 0)

    Points of inflection: -3, -1

    My textbook says the answer is
    concave up (- infinity , -3) , ( -1, inifinity)
    Concave down ( -3 , -1)
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  2. #2
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    Quote Originally Posted by Mudd_101 View Post
    Hey

    I don't understand how to determine the domain of concave up versus concave down when the graph has two points of inflection.

    For example

    x^4 + 8x^3 + 18x^2 + 8

    extreme points: -3, 0
    (graph is high at -3 and low at 0)

    Points of inflection: -3, -1

    My textbook says the answer is
    concave up (- infinity , -3) , ( -1, inifinity)
    Concave down ( -3 , -1)
    $\displaystyle f(x) = x^4 + 8x^3 + 18x^2 + 8$

    $\displaystyle f'(x) = 4x^3 + 24x^2 + 36x$

    $\displaystyle f'(x) = 4(x^3 + 6x^2 + 9x)$

    $\displaystyle f''(x) = 4(3x^2 + 12x + 9) = 12(x^2 + 4x + 3)$

    set $\displaystyle f''(x) = 0$ ...

    $\displaystyle 12(x + 3)(x + 1) = 0$

    $\displaystyle x = -3$, $\displaystyle x = -1$

    for $\displaystyle x < -3$, $\displaystyle f''(x) > 0$ ... $\displaystyle f(x)$ is concave up

    for $\displaystyle -3 < x < -1$, $\displaystyle f''(x) < 0$ ... $\displaystyle f(x)$ is concave down

    for $\displaystyle x > -1$ , $\displaystyle f''(x)> 0$ ... $\displaystyle f(x)$ is concave up
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