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how do you find the inflection points of a polynomial such as f(x)=12x^5+45x^4–80x^3+2 .

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    Look for points where the second derivative is zero. The first derivative (slope) must also change sign there.

    You will have to solve this cubic equation to get the inflection point.

    240 x^3 + 540 x^2 - 480 x = 0

    You can factor this down to
    x (12x^2 + 27 x -20) = 0

    x = 0 is one of the inflection points; there appear to be two others but the quadratic term does not factor easily.

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