Question

How do you find the inflection point of a cubic function?

Answer 1

Hint: First understand the meaning of the term ‘point of inflection’. We will take an example of any cubic equation $f\left( x \right)$. To find the point of inflection we will double differentiate the function \[f\left( x \right)\] to find the function \[f''\left( x \right)\]. We will substitute \[f''\left( x \right)\] equal to 0 and find the values of x. The values of x obtained will be the points of inflection.

Complete step by step answer:
Here, we have been asked to determine the points of inflection of a cubic function. But first we need to understand the meaning of the term ‘point of inflection’.
In differential calculus, the point of inflection or inflection point is a point on a smooth curve at which the curvature sign changes. If we will consider the graph of a function then we can say that the point of inflection is a point where the function changes from being concave to convex or from being convex to concave. For a double differentiable function, to find the point of inflection we use the condition \[f''\left( x \right)=0\], where \[f\left( x \right)\] is the given function.
Let us come to the question. We haven’t been provided with any particular function but it is only said to find the point of inflection for a cubic function. Let us take an example, say $f\left( x \right)={{x}^{3}}-12{{x}^{2}}+5x+8$. Now, differentiating both the sides with respect to x we get,
\[\Rightarrow f'\left( x \right)=3{{x}^{2}}-24x+5\]
Again differentiating both the sides with respect to x, we get,
\[\Rightarrow f''\left( x \right)=6x-24\]
Substituting \[f''\left( x \right)=0\] we get,
\[\begin{align}
  & \Rightarrow 6x-24=0 \\
 & \Rightarrow 6x=24 \\
 & \therefore x=4 \\
\end{align}\]

Hence, x = 4 is the point of inflection.

Note: Remember that \[f''\left( x \right)=0\] is not a sufficient condition for having a point of inflection. You must know about the points like: - stationary point, saddle point, critical point in differential calculus. There are many functions for which you will get \[f''\left( x \right)=0\] at point x = 0 but it will not be an inflection point. These things will be read in higher mathematics.