Find the maximum and minimum values of a trigonometry function
Question:
Given function f(x) = cos2x - 5 cos x + 6, find its maximum and minimum values.
Solution:
- let variable t = cos x
- because cos x is in the range of [-1, 1]
- so, variable t is in the range of [-1, 1]
- f(t) = t2 - 5t + 6
- = t2 - 5t + (5/2)2 - (5/2)2 + 6
- = [t - (5/2)]2 - 25/4 + 24/4
- = [t - (5/2)]2 - 1/4
Because the variable t is in the range of [-1, 1], when t = 1, the function has minimum value.
- fmin = f(1) = [1 - (5/2)]2 - 1/4
- = (-3/2)2 - 1/4
- = 9/4 - 1/4
- = 8/4 = 2.
When the variable t = -1, the function has maximum value.
- fmax = f(-1) = [-1 - (5/2)]2 - 1/4
- = (-7/2)2 - 1/4
- = 49/4 - 1/4
- = 48/4 = 12
Therefore, the function has minimum value 2 and the maximum value 12. Please watch the video for more details.