Question

How many turning points can a cubic function have?

Answer 1

Hint: A turning point is a point on a graph where it switches from increasing to decreasing (rising to falling) or decreasing to increasing (rising to increasing) (falling to rising). Or in other words, those points where the function attains its local extrema (maxima or minima) are known as turning points.

Complete step by step solution:
The turning points of a function are those where the function has a local maximum or local minimum. This implies that the function is increasing on one side of the turning point and decreasing on the other side. We know that the tangent at a point is function if the function is increasing at that place and negative if the function is decreasing. Now, since at turning points the function changes its nature as an increasing function to decreasing or vice-versa, we can conclude that the derivative of the function also changes its sign. So, we can say that at a turning point the derivative of the function is equal to zero. That being said, now let us look at the cubic function and use this result to find the possible number of points.
Graph of a cubic function with 2 turning points are marked below:

A cubic function is a polynomial of the form $a{x^3} + b{x^2} + cx + d$. We know that polynomials are differentiable on the entire real line. So, the cubic function is differentiable at all real points. You can find the derivative of this general form of cubic function to be $3a{x^2} + 2bx + c$, which is again a polynomial of degree 2. From the previous discussion we are aware that at turning points the derivative is zero. So, the turning points of this general cubic function are the roots of the equation $3a{x^2} + 2bx + c = 0$. Since, it is a quadratic equation it can have at most two distinct roots. So, the total possible count of points where the derivative is zero is 2. Hence, the cubic function can have at most two roots.

Note:
Note that the maximum possible number of turning points is two, but there could be a smaller number of points as well. In case of the function ${x^3}$, even though at $x = 0$ the derivative of the function $3{x^2}$ is zero, it is not a turning point as the sign of the derivative does not change at $x = 0$(It is positive in the entire real line). In fact, this function does not have a turning point.