Answer:
The roots for [tex]f(x) = \cos 4x - 4\cdot x^{2} + 9\cdot x[/tex] are [tex]x_{1} = -0.098[/tex] and [tex]x_{2} = 2.166[/tex], respectively.
Step-by-step explanation:
A root is a value of [tex]x[/tex] so that [tex]f(x) = 0[/tex]. Let suppose that function is the consequence of the subtraction between two functions, that is:
[tex]f(x) = g(x) - h(x)[/tex] (1)
If we know that [tex]f(x) = 0[/tex], [tex]g(x) = \cos 4 x[/tex] and [tex]h(x) = 4\cdot x^{2}-9\cdot x[/tex], then we have the following identity:
[tex]g(x) = h(x)[/tex]
We can estimate graphically the roots of [tex]f(x)[/tex] by graphing the following system:
[tex]y = \cos 4x[/tex] (2)
[tex]y = 4\cdot x^{2}-9\cdot x[/tex] (3)
Where roots are the points in which functions find each other.
With the help of a graphing, we estimate two solutions:
[tex](x_{1}, y_{1}) = (-0.098, 0.924)[/tex], [tex](x_{2}, y_{2}) = (2.166, -0.725)[/tex]
