show that the equation 2x-1-sinx=0 has exactly one real root.
sin x = 2x-1
2x = sin x - 1
range of sin x is -1 to + 1
but how do you know that 2x is between -2 and 0 and so x is between -1 and 0
If you view graphically the two equations
f1(x)=sin(x) ....(1)
f2(x)=2x-1 ....(2)
it will be obvious that there is only one real root.
To prove that there is only one real root, you could:
1. determine the interval I of x on which f2(x) is between -1 and 1.
2. find the root x0 where f1(x)=f2(x).
3. analyse the sign of the function f2(x)-f1(x) on the interval I on each side of x0.
Think graphically. The curve of sinx -1 oscillates between -2 and 0, touching zero at integer multiples of x = pi. It is -2 at x = 0.
The curve y = 2x can only intersect sinx -1 at one point, somewhere between x = -pi and 0. For x > 0 and x < -pi, it is too large to ever intersect sinx -1 again.
The curve of sinx -1 oscillates between -2 and 0, touching zero at ODD integer multiples of x = pi. The argument remains the same for the reason there is only one root.
To show that the equation 2x - 1 - sin(x) = 0 has exactly one real root, we need to find a way to prove that it intersects the x-axis at only one point.
To solve this equation, we can start by rearranging it as follows:
2x - 1 = sin(x)
By observing the range of the sine function, which is between -1 and +1, we know that sin(x) can take values between -1 and +1.
Therefore, if 2x - 1 = sin(x), it implies that 2x must also be between -2 and 0. This is because if sin(x) is between -1 and +1, subtracting 1 from it yields a range of -2 to 0.
So, we have: -2 < 2x - 1 < 0
To find the range of x, we divide each part of the inequality by 2:
-1 < x - 0.5 < 0
Adding 0.5 to each part of the inequality, we get:
-0.5 < x < 0.5
Therefore, we have determined the range of x based on the given equation 2x - 1 - sin(x) = 0. The solution to this equation is x between -0.5 and 0.5, meaning there is exactly one real root.
To summarize:
The equation 2x - 1 - sin(x) = 0 has exactly one real root because the range of x satisfying the equation is between -0.5 and 0.5, resulting in a single point of intersection with the x-axis.