Help Solving x - sinx for -pi/2 to pi/2

  • Thread starter mohlam12
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In summary, the conversation discusses the function a(x) = x - sin x and the need to demonstrate that for x>=0, a(x)>=0. The individual has tried to derive the function, but is facing difficulties when it comes to demonstrating the same thing for x<0. The expert notes that for x<0, a(x)<=0 and explains that simply stating the derivative is positive is not enough to prove that a(x) is positive. The trivial fact that sin(0)=0 must also be taken into account.
  • #1
mohlam12
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hey everybody
i need help solving this.. i have e function a(x) = x - sin x x belongs to -pi/2, pi/2

i need to demonstrate that for x>=0, a(x)>=0

i tried to derivate the function to get 1-cosx which is positive, therefore a(x) is positive too.. but then i'd face problems when it comes to demonstrating the same thing for x<0

any hints ?!

thanks
 
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  • #2
It's good that you "face problems when it comes to demonstrating the same thing for x< 0"! The "same thing", that a(x)>= 0, is not true for x< 0. If x< 0, then with u= -y, a(y)= -y- sin(-y)= -y+ sin(y)= -(y- sin(y)). Since y> 0, y- sin(y)> 0 so -(y- sin(y))< 0. For x<= 0, a(x)<= 0.

By the way, you can't simply say "1- cos(x) is positive, therefore a(x) is positive too". The fact that the derivative is positive tells you that x- sin(x) is increasing. You have to add the trivial fact that sin(0)= 0 in order to argue that it is then positive.
 

1. What is the solution to x - sinx for -pi/2 to pi/2?

The solution to x - sinx for -pi/2 to pi/2 is a continuous function that can be represented by the graph of y = x - sinx. It has many solutions, which can be found by using numerical methods or by graphing the function.

2. Is there a closed-form solution to x - sinx for -pi/2 to pi/2?

No, there is no closed-form solution to x - sinx for -pi/2 to pi/2. This means that the solutions cannot be expressed using a finite combination of elementary functions.

3. How do I solve x - sinx for a specific value of x?

To solve x - sinx for a specific value of x, you can plug in the value for x into the equation and calculate the corresponding value for sinx. You can also use a graphing calculator or online tool to find the solution.

4. What is the maximum and minimum value of x - sinx for -pi/2 to pi/2?

The maximum value of x - sinx for -pi/2 to pi/2 is pi/2, which occurs at x = pi/2. The minimum value is -pi/2, which occurs at x = -pi/2.

5. How does changing the interval of x affect the solutions to x - sinx?

Changing the interval of x will change the range of solutions for x - sinx. For example, if the interval is changed to -pi to pi, the solutions will include all real numbers. On the other hand, if the interval is changed to -2pi to 2pi, the solutions will repeat every 2pi. The shape of the graph will also change accordingly.

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