A random point $(a,b)$ is uniformly distributed in a unit square $K=[(u,v):0<u<1,0<v<1]$

Question

A random point $(a,b)$ is uniformly distributed in a unit square $K=[(u,v):0<u<1,0<v<1]$ Let $n$ be the number of real roots of the polynomial $f(x)=\frac{1}{3}x^3 - a^2 x +b$.Find the probability $P(n=3)$. I am failing to understand the criteria of choosing $(a,b)$ so that $f(x)$ has three real roots.

Answer 1

The graph of $f$ is a cubic parabola. We have three real roots of $f(x)=0$ iff $f$ has a local maximum and a local minimum, and if the values of $f$ at these two points have different sign. From $$f'(x)=x^2-a^2$$ we can conclude that the two local extrema exist at $\pm a$ as required. Furthermore $$f(a)f(-a)=\left(b-{2\over3}a^3\right)\left(b+{2\over3}a^3\right)\ .$$ You now have to analyze how the sign condition for this product partitions the unit square $K$.The probability you are after is the area of one part of this partition.