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# Find the maximum and minimum values of a trigonometry function

Question:

Given function f(x) = cos^{2}x - 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) = t
^{2}- 5t + 6 - = t
^{2}- 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.

- f
_{min}= 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.

- f
_{max}= 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.