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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.