Given:
The roller-coster is moving in the trajectory of this curve
[tex]f(x)=3x^4-18x^3-21x^2+144x-108[/tex]
Step by step solution:
To solve this complete problem we need to draw the estimated graph of this function, so that we can answer this question easily.
First of all, we need to find the roots of the given equation,to plot the curve:
let us put the random numbers that may satisfy the equation:
Let us put x = 1:
[tex]\begin{gathered} f(x)=3x^4-18x^3-21x^2+144x-108 \\ \\ f(1)=3-18-21+144-108 \\ \\ f(1)=\text{ 0} \end{gathered}[/tex]
From here we can say that 1 is the root of the equation.
We will now divide this function from (x-1), so that we can get the cubic equation:
We will use long division method for division, the result we get after the division is:
[tex]f(x)=(x-1)(3x^3-15x^2-36x+108)[/tex]
We will now try to factorize the cubic equations, by putting the random numbers that may satisfy the equation:
let us put x = 2:
[tex]\begin{gathered} f(x)=(x-1)(3x^3-15x^2-36x+108) \\ \\ f(2)=(2-1)(3(2)^3-15(2)^2-36(2)+108) \\ \\ f(2)=(1)(24\text{ }-\text{ 60 - 72 +108}) \\ \\ f(2)=0 \end{gathered}[/tex]
From here we can say that f(2) is also the root of this cubic
We will now divide the cubic equation with (x-2), so we can break the cubic into quadratic:
Upon division the cubic equation break into following factors:
[tex]\begin{gathered} =(x-2)(3x^2-9x-54) \\ \\ which\text{ further simplified into:} \\ \\ =(x-2)(x-6)(x+3) \end{gathered}[/tex]
From here we have found out four roots of the initial function that are:
x = 1,2,6,-3
Now we can easily plot the curve:
This is estimated curve, there are no sharp edges.
On the basis of this curve, we can easily answer all the questions related to this curve.
