find the difference between the greatest and least value of function cos x+1/2cos2x-1/3cos3x
9/4
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To find the difference between the greatest and least value of a function, we need to determine the maximum and minimum values of the function. In this case, the function is given by:
f(x) = cos(x) + (1/2)cos(2x) - (1/3)cos(3x)
To find the maximum and minimum values, we can take the derivative of the function and find the critical points. Set the derivative equal to zero and solve for x to find the critical points.
Step 1: Find the derivative of f(x):
f'(x) = -sin(x) - sin(2x) + sin(3x)
Step 2: Set f'(x) = 0 and solve for x:
-sin(x) - sin(2x) + sin(3x) = 0
Solving this equation may be challenging due to the combination of trigonometric functions. You can simplify it by using trigonometric identities or by using technology such as a graphing calculator or software.
Step 3: Once you have found the critical points, plug these values of x into the original function f(x) to find the corresponding y-values. This will give you a list of values of f(x).
Step 4: Determine the greatest and least values from the list of f(x). The difference between the greatest and least values will be the result you are looking for.