Taylor polynomial, approximative solution of this equation

[Kqwert]

Homework Statement

The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0.

Homework Equations

The Attempt at a Solution

So as I understand the task we should find a Taylor polynomial of (1/3)*cos(3x) around 0. I have found this to be
(1/3)-(3x^2)/2. However, which value of x should I put into the equation in order to estimate the solution? I know it must be in the interval [0,1], but unsure of the value.

 

[FactChecker]

Subtract the 4x from both sides and then you are looking for the zeros of a second order polynomial. Use the quadratic equation to find the values of x that give zero.

 

[Kqwert]

Excellent, thank you!

 

[Ray Vickson]

Homework Statement

The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0.

Homework Equations

The Attempt at a Solution

So as I understand the task we should find a Taylor polynomial of (1/3)*cos(3x) around 0. I have found this to be
(1/3)-(3x^2)/2. However, which value of x should I put into the equation in order to estimate the solution? I know it must be in the interval [0,1], but unsure of the value.

You have an equation of the form $x = f(x)$ (where $f(x) = (1/12) \cos(3x)$), and you replace it by a simpler approximate equation $x = g(x),$, where $g(x)$ is an approximation to $f(x).$ Just go ahead and solve the simpler, approximate equation.