# Real roots and coefficients for a fourth degree polynomial

Calculus Level 4

Let $$P(x)=\frac{1}{4}x^{4}+\frac{1}{3}x^{3}+ax+b$$ be a polynomial that has a negative value at a certain point. A pair of number $$(a, b)$$ satisfies that the polynomial $$P(x)$$ has only two different real roots and no complex roots, and the product $$ab$$ has the maximum possible value. Find $$\frac{a}{b}.$$

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