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Math Help - concave polynomials

  1. #1
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    concave polynomials

    Give an example of a polynomial that is concave up on R and whose second derivative has at least two distinct roots.
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  2. #2
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    Lets start with what you want.

    To be concave up the 2nd derivative must be greater than zero.

    f''(x) \ge 0 and we want it to have two distinct real roots.

    The only way it can have roots and not cross the x-axis is if they are repeated and of even degree.

    Then for any real numbers r_1 \ne r_2

    f''(x)=(x-r_1)^2(x-r_2)^2

    This satisfies the above now you just need to find f.

    This should get you started.
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