Question

How many stationary points can a cubic function have?

Answer 1

Hint: As we know that a cubic function is a function of the form, where the coefficients $a,b,c$ and $d$ are the real numbers and the variable $x$ takes real values and $a \ne 0$. Or we can say it is a polynomial of degree there. The standard form of a cubic function can be written as $a{x^3} + b{x^2} + cx + d.$ The basic cubic function is $f(x) = {x^3}$.

Complete step by step solution:
We know that a stationary point of a function $f(x)$ is a point where the derivative of $f(x)$ is equal to $0$.
Let us consider the general equation of a cubic polynomial: $A{x^3} + B{x^2} + Cx + D\left\{ {A,B,C,D} \right\} \in R$.
Now consider $f'(x) = 3A{x^3} + 2Bx + C$, the stationary points of $f(x)$ will be where $f'(x) = 0$.
We know that $f'(x)$ is a quadratic function, so it will have $2$ real or complex roots.
Since there are two real or complex roots of $f'(x)$ where it will be zero, so the function $f(x)$ will have two most real stationary points.
Hence a cubic polynomial with real coefficients can have at most $2$ real stationary points.

Note: We should note that these points are called stationary points because at these points the function is neither increasing nor decreasing. In order to evaluate the stationary points of a function that we have to consider $y = f(x)$, so here the stationary points will be given by $\dfrac{{dy}}{{dx}} = 0$. The highest power over the variable $x$ is $3$ in a cubic polynomial.