Question

Find the extreme values of cos 2x +cos^2 x

Answer 1

Answer:

Maximum 2 when x = kπ.

Minimum -1 when x = (k + 1/2)π

Step-by-step explanation:

f(x) = cos 2x + cos² x

= ( 2 cos² x - 1 ) + cos² x

= 3 cos² x - 1

As -1 ≤ cos x ≤ 1, we have 0 ≤ cos² x ≤ 1

Maximum values

The maximum value of f(x) is when cos² x = 1, which makes f(x) = 3-1 = 2.

This occurs when cos x = ±1, so when x = kπ, for any integer k.

Minimum values

The minimum value of f(x) is when cos² x = 0, which makes f(x) = 0-1 = -1.

This occurs when cos x = 0, so when x = (k + 1/2)π, for any integer k.