Solving for x involving trig

[kpou]

I have formula z(x)=2ASin(7*pi*x)Cos(3*pi*x)
I need to find the location where the displacement is max and min.

I understand how to get the differential z'(x)=A4piCos(4*pi*x)+A10piCos(10*pi*x)
Then I need to find critical points... this is where I have been having the trouble. My calculator keeps telling me my dimensions don't work, and it is really killing me. How do you solve for that = 0?

Edit: z(x) can also be written as A(Sin(4*pi*x)+Sin(10*pi*x)) if that saves you any work
Also: x is between 0 and 1meter

-----------Problem Solved----------

 

[lippyka]

The solution to finding the maximum and minimum displacement is to set the derivative of the function equal to 0 and solve for x.z'(x) = A4piCos(4*pi*x)+A10piCos(10*pi*x) = 0This equation can be simplified by dividing both sides by A:4piCos(4*pi*x)+10piCos(10*pi*x) = 0To solve this equation, we use the quadratic formula: x = [-b ± √(b2 - 4ac)]/2aWhere a = 4pi, b = 10pi and c = 0.So, x = [(-10pi) ± √(102pi2 - 0)]/2(4pi)x = [-10pi ± 0]/8pix = -1.25 or x = 0.25Since x is between 0 and 1 meter, the only valid solution is x = 0.25. Therefore, the location where the displacement is a maximum is x = 0.25 meter. At this point, z(0.25) = A(Sin(π) + Sin(5π)) = 2A. In order to find the minimum displacement, we need to find the local minimums of the function z(x). This can be done by setting the second derivative of the function equal to 0 and solving for x.z''(x) = A(-4π2Sin(4πx)+10π2Sin(10πx)) = 0We can rearrange this equation to get:Sin(4πx)=Sin(10πx)This equation has an infinite number of solutions; however, since x must be between 0 and 1 meter, the only valid solution is x = 0.75. Therefore, the location where the displacement is a minimum is x = 0.75 meter. At this point, z(0.75) = A(Sin(3π) + Sin(7π)) = -2A. Therefore, the maximum displacement is 2A and the minimum displacement is -2A.