Question

Find all the roots of the given function. Use preliminary analysis and graphing to find good initial approximations. ​f(x)equals=cosine left parenthesis 3 x right parenthesis minus 7 x squared plus 4 xcos(3x)−7x2+4x

Answer 1

The given function is

f(x)=cos 3x-7 x²+ 4x

f'(x)=-3 sin 3 x-14 x+4

When you will draw the graph of the function , you will find that root of the function lie between (-1,0).

Consider initial root as,

`x_(0)=0`

Using Newton method to find the roots of the equation

`x_(n+1)=x_(n)-\frac{f{x_n}}{f'{x_(n)}}\\\\x_(1)=x_(0) - (cos 3x_(0)-7 x_(0)^2+ 4x_(0))/(-3 sin 3 x_(0)-14 x_(0)+4)\\\\x_(1)=-(\cos 0^(\circ)-0+0)/(-3 * 0-0+4)\\\\x_(1)=(-1)/(4)\\\\x_(1)= -0.25\\\\x_(2)=x_(1) - (cos 3x_(1)-7 x_(1)^2+ 4x_(1))/(-3 sin 3 x_(1)-14 x_(1)+4)\\\\x_(2)=-0.25 -(cos (-0.75)-7* (0.0625)- 1)/(-3 sin (-0.75)+3.50+4)\\\\x_(2)= -0.176054`

`x_(3)=x_(2) - (cos 3x_(2)-7 x_(2)^2+ 4x_(2))/(-3 sin 3 x_(2)-14 x_(2)+4)\\\\x_(3)=-0.176054 -(cos (3* -0.176054)-7* (-0.176054)^2+4 * -0.176054)/(-3 sin (-0.176054)-14 * (-0.176054)+4)\\\\x_(3)= -0.1689`

So, root of the equation is

=0.1688878

=0.1689(approx)