Prove that $(x-1)^3+(x-2)^3+(x-3)^3+(x-4)^3=0$ has only one real root.
It's easy to show that the equation has a real root using Rolle's theorem. But how to show that the real root is unique? By Descartes' rule of sign, it can be shown that it has 3 or 1 real root.
But it doesn't guarantee that the real root is unique. If we calculate the root then it can be shown that it has only one real root.
Best Answer
The function is strictly increasing so the function is one to one.