I have a question which asks: At what value of b does the quadratic 1.5x^2+bx+2.34 move from having a stable fixed point to having a stable period 2 point? How would I go about solving this?
2026-05-10 20:03:49.1778443429
Value of b which moves quadratic from a stable fixed point to a stable period 2 point
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It sounds like you are iterating $x_{i+1}=1.5x_i^2+bx_i+2.34$ You can find the fixed point by solving $x_{i+1}=x_i$, giving the quadratic $x=1.5x^2+bx+2.34$. The fixed point will be a function of $b$. It is stable if the derivative of $1.5x^2+bx+2.34$ evaluated at the fixed point has absolute value less than $1$. This is because the slope of the curve is the factor the error is multiplied by each step. If the absolute value is less than 1, you get closer. If the absolute value is greater than 1, you get farther away.