There are some important deductions to determine the shape of the cubic curve using determinant & the nature of at least one real root of any cubic equation simply by observing the value of constant. These are very useful in plotting the graphs of cubic functions.

1. Mr Harish Chandra Rajpoot
M.M.M. University of Technology, Gorakhpur-273010 (UP), India 16/11/2014
General form of the cubic equation:
We know that a cubic equation has either one or three real roots.
Here, we are interested to study the cubic equations having for ease of understanding. We usually take because can be made positive in any cubic equation by multiplying it either by or by & the curve of ( ) is simply obtained by reflecting the curve of ( ) about the x-axis.
Nature of at least one real root: We can easily determine the nature of at least one real root simply by observing the sign of constant in any cubic equation “The nature of at least one real root of any cubic equation is always opposite to the sign of constant .” 1. ⇒ 2. ⇒ 3. ⇒
Proof: Nature of Cubic Curve ( )
The nature (shape) of the cubic curve is the proof of the above statements. In order to find the nature of cubic curve, let’s find out the conditions for maxima & minima as follows ⇒ ( ) ( ) ( ) ( )
Now, for local maxima or minima, ( ) ⇒ ⇒ √( ) ( )( ) ( ) √ √ √ ( )
Now, let’s check out the local maxima & minima for above values of variable as follows
Case 1: If ⇒ i.e. the points of local maxima & minima will always be real & distinct. ⇒ ( ) ( √ )

2. √ ( )
Hence, in this case, the cubic curve ( ) has local maxima at ⇒ ( ) ( √ ) √ ( )
Hence, in this case, the cubic curve ( ) has local minima at
The graph of cubic curve ( ) will always be of Wavy Nature i.e. cubic curve has both local maxima & minima successively at two distinct points . As shown in the figure 1 below
In this case, the cubic equation ( ) has either one or three real roots. But at least one real root of equation has sign opposite to the sign of constant .
Case 2: If ⇒ i.e. the points of local maxima & minima will be equal (same) as follows ⇒ ( ) ( ) ( ) ( ) ( )
Neither maxima nor minima
Hence, in this case, the cubic curve ( ) has neither local maxima nor local minima. It is called as strict nature of cubic curve (as shown in the figure 2 below). In this case, the cubic equation ( ) has only one real distinct root &
a. Curve either crosses the x-axis (only one real root with sign opposite to that of constant d) or
b. It is tangent to the x-axis (i.e. the cubic equation has three equal real roots or one distinct real root). Figure 1: Cubic curve successively has one local maxima at lower value of x & one local minima at higher value of x i.e. it has wavy nature

3. Case 3: If ⇒ i.e. the cubic curve has no point of local maxima & minima as shown in the figure 2 below.
Figure 2: Cubic curve has no local maxima or minima. It has one distinct real root either by crossing the x-axis or being tangent to the x-axis (i.e. three equal real roots). This nature of curve is called strict
Similar to the case 2, the cubic curve ( ) has neither local maxima nor local minima. It is called as strict nature of the cubic curve (as shown in the figure 2 above). In this case, the cubic equation ( ) has only one real distinct root &
a. Curve either crosses the x-axis (only one real root with sign opposite to that of constant d) or
b. It is tangent to the x-axis (i.e. the cubic equation has three equal real roots).
It is clear from above cases that for any position of the origin ( )& for any values of coefficient, the nature of at least one real root must always be opposite to sign of the constant in the cubic equation.
Conclusion: From above three cases, it can be concluded that a cubic may have
1. Either Strict Nature (I.e. cubic curve has no local maxima or minima) or
2. Wavy Nature (i.e. cubic curve has both local maxima & minima) The nature of any cubic curve is of two types which is determined by using the determinant (D)
Case 1: ⇒
In this case, the cubic equation ( ) has either one or three real roots. But at least one real root has sign opposite to the sign of constant . See the figure 1 above. Any cubic equation can’t have single maxima or single minima i.e. if a cubic has local maxima then it definitely has local minima at some distinct point & vice-versa is true. Thus, local maxima & local minima are always present as a pair in a cubic equation & single maxima/minima can’t exist in any cubic equation.
Case 2: ⇒

4. In this case, the cubic equation ( ) has only one distinct real root &
a. Curve either crosses the x-axis (only one real root with sign opposite to that of constant d) or
b. It is tangent to the x-axis (i.e. cubic equation has three equal real roots). See the figure 2 above In any cubic equation , the local maxima is always followed by the local minima if any.
The cubic equation is easily studied by multiplying it by to get then using the above deductions for determining the nature of one real root & the curve ( ) is obtained by reflecting the curve of ( ) about the x-axis.
Types of cubic equation on the basis of values of coefficients : There are types of the cubic equation as tabulated below
S/No.
( )
( )
( )
( )
Nature of at least one real root of cubic equation
(opposite to the sign of ) 1 Positive
2
Zero 3 Negative
4
Positive 5 Zero
6
Negative 7 Positive
8
Zero 9 Negative
10
Positive 11 Zero
12
Negative 13 Positive
14
Zero 15 Negative
16
Positive 17 Zero
18
Negative 19 Positive
20
Zero 21 Negative
22
Positive 23 Zero
24
Negative 25 Positive
26
Zero 27 Negative
Similarly, we can obtain other 27 types of cubic equations for but these are the transformed forms (i.e. reflections about the x-axis) of above 27 types of the cubic equation. So we are not considering those.
Note: Above articles had been concluded & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)
M.M.M. University of Technology, Gorakhpur-273010 (UP) India Nov, 2014
Email: rajpootharishchandra@gmail.com
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot