1)sinx=x  x=? 2)2cosx=x x=? 3) f'(x)=0= (x-cosx)'   x=?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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