How to prove that a particular cubic equation has three real and distinct roots without finding its discriminant via calculus method?
Please do not use mathematical concepts beyond high school/IIT-JEE. Well and if someone is using the method of finding stationary points please explain the logic behind the technique.
I suppose you know derivatives.
Given $y(x)=ax^3+bx^2+cx+d$, find the derivative $ y'(x)=3ax^2+2bx+c$.
To find stationary points you have to solve the equation: $$ 3ax^2+2bx+c=0 $$ that is second degree. So: if this equation has no real roots then the cubic has only one real root. If you find two real solutions $x_1,x_2$ then:
if $ y(x_1)y(x_2) <0$ the cubic has three distinct real roots,
if $ y(x_1)y(x_2) =0$ the cubic has two distinct real roots, one of them double, or three coincident real roots.
if $ y(x_1)y(x_2) >0$ the cubic has only one real root.