Question

How can we solve questions of Continuity Differentiability involving cubic equations?

Answer 1

Continuity and differentiability and cubic equations are 2 separate topics. It is possible that a probem in continuity or differentiability may lead to a cubic equation.
The simplest way to solve a cubic equationis to find 1 root using factor thorem or remainder theorem through trial and error. If (x-a) is a root, then P(a) = 0 where P(x) = 0 is the cubic equation, Once you have found 1 root, you can divide the polynomial by this (x-a) to get a quadratic. The quadratic will either have no root or 2 roots (sometimes equal). For example
If we have the cubic equation 2x³ + 3x² – 11x – 6 = 0, we can find by trial and error that substituting x = 2 gives a zero. So we divide by x-2 to get $2x^2+7x+3$ which can be further factored as (2x+1)(x+3).
So the complete solution is x = 2, -1/2 or -3.