Question

Find all the roots of the given function on the given interval. Use pre-liminary analysis and graphing to find good initial approximations.

f(x) = cos (4x) - 4x2 + 9x

Answer 1

The roots for
`f(x) = \cos 4x - 4\cdot x^(2) + 9\cdot x` are
`x_(1) = -0.098` and
`x_(2) = 2.166`, respectively.

Explanation:

A root is a value of
`x` so that
`f(x) = 0`. Let suppose that function is the consequence of the subtraction between two functions, that is:

`f(x) = g(x) - h(x)` (1)

If we know that
`f(x) = 0`,
`g(x) = \cos 4 x` and
`h(x) = 4\cdot x^(2)-9\cdot x`, then we have the following identity:

`g(x) = h(x)`

We can estimate graphically the roots of
`f(x)` by graphing the following system:

`y = \cos 4x` (2)

`y = 4\cdot x^(2)-9\cdot x` (3)

Where roots are the points in which functions find each other.

With the help of a graphing, we estimate two solutions:

`(x_(1), y_(1)) = (-0.098, 0.924)`,
`(x_(2), y_(2)) = (2.166, -0.725)`