Question

How do you find the inflection points of the graph of the function: $f(x) = 2x{(x - 4)^3}$

Answer 1

Hint: Second derivation of any kind of graph function shows inflection point.
When we do a second derivation of the graph function to see for the change in the concavity nature of the graph like change from concave up to concave down or vice versa. The point at which the change starts is called the inflection point.

The inflection points are given when the second derivative is equalled to zero. Because the second derivative indicates the change in the concavity of the graph function in the question.

Complete step by step answer:
From the question, we can see that the given equation is
$f(x) = 2x{(x - 4)^3}$
So, we first begin by doing the first derivative of the function, and then proceed to the second derivation.
FIRST DERIVATION: -
$\dfrac{{dy}}{{dx}} = 2{(x - 4)^3} + 2x(3{(x - 4)^2})$
Then we further simplify the derived equation, as following
$
  \dfrac{{dy}}{{dx}} = 2{(x - 4)^3} + 6x{(x - 4)^2} \\
  \dfrac{{dy}}{{dx}} = [{(x - 4)^2}][(2x - 8) + (6x)] \\
  \dfrac{{dy}}{{dx}} = (8x - 8){(x - 4)^2} \\
 $
Here, we have obtained a simplified form of the equation.
Now, we have to do second derivation of the equation
We get;
$\dfrac{{{d^2}y}}{{d{x^2}}} = 8{(x - 4)^2} + (8x - 8)(2x - 8)$

We used the ${(a + b)^2}$ formula and expanded and did the derivation.
On further simplification, we get
$
  \dfrac{{{d^2}y}}{{d{x^2}}} = 8({x^2} - 8x + 16) + (16{x^2} - 80x + 64) \\
  \dfrac{{{d^2}y}}{{d{x^2}}} = 24{x^2} - 144x + 192 \\
$
For the inflection point, we need to know the point of change of the concavity nature of the function.
$24({x^2} - 6x + 8) = 0$

We have a quadratic equation and we need to find the roots of it which are going to be the x-coordinates of the inflection point.
$
   \Rightarrow \dfrac{{6 \pm \sqrt {36 - (4)(1)(8)} }}{2} \\
   \Rightarrow \dfrac{{6 \pm \sqrt 4 }}{2} \\
   \Rightarrow \dfrac{{6 \pm 2}}{2} \\
$

We get two coordinates here, which are
X1= 4
X2= 2
And on substituting these values of x found in the second derivative into the original function, we will obtain the inflection points.
$
  f(4) = 2(4){(4 - 4)^3} = 0 \\
  f(2) = 2(2){(2 - 4)^3} = 4.( - 8) = - 32 \\
$

So, finally the inflection points of the given function are
$A(4,0)B(2, - 32)$
Below is the graphical representation of the given equation with the inflection points marked.

Note: We have to be aware that inflection points are not any random points but rather they show the change in the nature of concavity of the curve from one form to another form, nor these are the start or endpoint of the curve.