Question

Find the extreme value of cos 2x + cos² x.

Answer 1

we know, cos²A = (1 + cos2A)/2

so, cos²x = (1 + cos2x)/2

now, cos2x + cos²x = cos2x + (1 + cos2x)/2

= (2cos2x + 1 + cos2x)/2

= (1 + 3cos2x)/2

we know, -1 ≤ cosine function ≤ 1

so, -1 ≤ cos2x ≤ 1

or, -3 ≤ 3cos2x ≤ 3

or, 1 - 3 ≤ 1 + 3cos2x ≤ 1 + 3

or, -2 ≤ 1 + 3cos2x ≤ 4

or, -2/2 ≤ (1 + 3cos2x)/2 ≤ 4/2

or, -1 ≤ (1 + 3cos2x)/2 ≤ 2

hence, -1 ≤ cos2x + cos²x ≤ 2

so, maximum or extreme value of given function is 2.