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.

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.