1)sinx=x x=? 2)2cosx=x x=? 3) f'(x)=0= (x-cosx)' x=?
1) Since the given equation `sin x = x` is a transcendental equation, you should solve it for x use graphical method such that:
Notice that the graphs of the functions `y = sin ` x and `y = x` intersect at `x = 0` , hence, the solution to the given equation is `x = 0.`
2) The equation `2cos x = x` is a transcendental equation, hence, you need to use graphical method such that:
Notice that the graphs of functions `y = cos x` and `y = x/2` intersect at `x = 0` and a value of `x in (1,1.5)` , hence, the equation has two solutions.
3) You should differentiate the function `f(x) = x-cosx` with respect to x such that:
`f'(x) = 1 + sin x`
`f'(x) = 0 => 1 + sin x = 0 => sin x = -1 => x = (-1)^n*sin^(-1)(-1) + npi`
`x = (-1)^(n+1)*(pi/2) + npi`
Hence, evaluating the general solution to the equation `f'(x) = 0` yields `x = (-1)^(n+1)*(pi/2) + npi.`