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Math Help - 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)
    f(x) = x^4 + 8x^3 + 18x^2 + 8

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

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

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

    set f''(x) = 0 ...

    12(x + 3)(x + 1) = 0

    x = -3, x = -1

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

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

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