Question

How do you find the inflection points of the graph of the function: $y = \left( {\dfrac{1}{{{x^2}}}} \right) - \left( {\dfrac{1}{{{x^3}}}} \right)$?

Answer 1

We will first find the derivative of y and then find the derivative again so that we get the second derivative of y. Then, we will equate it to zero to find the inflection points.

Complete step by step solution:
We are given that we are required to find the inflection points of the graph of the function: $y = \left( {\dfrac{1}{{{x^2}}}} \right) - \left( {\dfrac{1}{{{x^3}}}} \right)$.
Now, let us find the derivative of both the sides in the above equation with respect to $x$, we will then obtain the following equation with us:-
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{1}{{{x^2}}}} \right) - \dfrac{d}{{dx}}\left( {\dfrac{1}{{{x^3}}}} \right)$
We can write this as follows:-
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{x^{ - 2}}} \right) - \dfrac{d}{{dx}}\left( {{x^{ - 3}}} \right)$
Now, solving the right hand side of the above equation, we will then obtain the following equation:-
$\Rightarrow \dfrac{{dy}}{{dx}} = - 2{x^{ - 3}} - \left( { - 3{x^{ - 4}}} \right)$
Simplifying the right hand side further, we will then obtain the following equation with us:-
$\Rightarrow \dfrac{{dy}}{{dx}} = - 2{x^{ - 3}} + 3{x^{ - 4}}$
Finding the derivative of both the sides of above equation again with respect to $x$, we will then obtain the following equation with us:-
$\Rightarrow \dfrac{d}{{dx}}\left( {\dfrac{{dy}}{{dx}}} \right) = \dfrac{d}{{dx}}\left( { - 2{x^{ - 3}}} \right) + \dfrac{d}{{dx}}\left( {3{x^{ - 4}}} \right)$
We can write this as follows:-
$\Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} = \dfrac{d}{{dx}}\left( { - 2{x^{ - 3}}} \right) + \dfrac{d}{{dx}}\left( {3{x^{ - 4}}} \right)$
Now, solving the right hand side of the above equation, we will then obtain the following equation:-
$\Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} = 6{x^{ - 4}} - 12{x^{ - 5}}$
We can write this equation as follows:-
$\Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} = \dfrac{6}{{{x^4}}} - \dfrac{{12}}{{{x^5}}}$
Now, substituting this equal to 0, we will then obtain the following equation:-
$\Rightarrow \dfrac{6}{{{x^4}}} - \dfrac{{12}}{{{x^5}}} = 0$
Simplifying it, we will then obtain the following equation with us:-
$\Rightarrow 6 \times {x^5} = 12 \times {x^4}$
Simplifying it further, we will then obtain the following equation with us:-
$\Rightarrow {x^4}\left( {x - 2} \right) = 0$

Therefore, x = 0 and 2 are the points of inflection.

Note:-
The students must note that the inflection points are derived by substituting the double derivative of the function equal to zero.
The students must also notice that we have used the following fact in the solution mentioned above:-
$\Rightarrow \dfrac{d}{{dx}}\left( {{x^{ - n}}} \right) = - n{x^{ - n - 1}}$
The students might also make the mistake of crossing – off the x raised to the power 4 from the last step but you must keep in mind the possibility of x being zero. Because if some variable is equal to 0, we can never cancel it off.