The equation of the quartic function is [tex]y = -\frac{1}{2}\cdot x^{4}+x^{3} +\frac{3}{2}\cdot x^{2} -2\cdot x - 2[/tex], whose factored form is [tex]y = (x-2)^{2}\cdot (x+1)^{2}[/tex].
A quartic function is a fourth order polynomial, whose form is presented below:
[tex]y = a\cdot x^{4}+b\cdot x^{3} + c\cdot x^{2}+d\cdot x + e[/tex], [tex]a, b, c, d, e\in \mathbb{R}[/tex] (1)
Where:
- [tex]x[/tex] - Independent variable.
- [tex]y[/tex] - Dependent variable.
Given that such polynomial must be tangent to the x-axis in two place, slope must be zero and by differential calculus we have the following expression for the slope of the tangent line:
[tex]4\cdot a \cdot x^{3} + 3\cdot b\cdot x^{2} + 2\cdot c \cdot x + d = 0[/tex] (2)
Then, we can construct the following system of linear equations:
[tex]e = -2[/tex] (3)
[tex]a -b +c -d = -e[/tex] (4)
[tex]-4\cdot a +3\cdot b -2\cdot c + d = 0[/tex] (5)
[tex]16\cdot a + 8\cdot b + 4\cdot c + 2\cdot d = -e[/tex] (6)
[tex]32\cdot a + 12\cdot b + 4\cdot c + d = 0[/tex] (7)
The solution of the system is: [tex]a = -\frac{1}{2} , b = 1, c = \frac{3}{2}, d = -2[/tex].
The equation of the quartic function is [tex]y = -\frac{1}{2}\cdot x^{4}+x^{3} +\frac{3}{2}\cdot x^{2} -2\cdot x - 2[/tex], whose factored form is [tex]y = (x-2)^{2}\cdot (x+1)^{2}[/tex].
