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Math Help - Show the cubic function has extreme value

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    Show the cubic function has extreme value

    Show that the cubic  p(x) = x^3 + ax^2 + bx + c has extreme values iff a^2 > 3b.
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    cubics

    Any cubic goes from -infinityto +infinity. The extreme values occur when there is a bump, i.e., the derivative is zero. When you take the derivative, you'll get a quadratic. If it has real roots, you've found where the bumps are. The quadratic formula should help you.
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    Quote Originally Posted by qmech View Post
    Any cubic goes from -infinityto +infinity. The extreme values occur when there is a bump, i.e., the derivative is zero. When you take the derivative, you'll get a quadratic. If it has real roots, you've found where the bumps are. The quadratic formula should help you.
    ok, so now I have p'(x) = 3x^2 + 2ax +b

    so do I set p'(x) = 0 and find the root? But how do I solve the equation?
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  4. #4
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    Use the quadratic formula. You know, that stuff with x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
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  5. #5
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    Quote Originally Posted by 450081592 View Post
    ok, so now I have p'(x) = 3x^2 + 2ax +b

    so do I set p'(x) = 0 and find the root? But how do I solve the equation?
    p'(x) needs to have two real roots ... the discriminant, b^2-4ac > 0

    (2a)^2 - 4(3)(b) > 0

    4a^2 - 12b > 0

    a^2 - 3b > 0

    a^2 > 3b
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