Find the Largest Value of the Function y = 2cos4x + √cos ^ 2 4x-1

We transform the given function depending on the sign of cos (4x):

y = 2cos (4x) + √cos ^ 2 (4x) – 1;
y = 2cos (4x) ± | cos (4x) | – one;
a) cos (4x) <0;

y = 2cos (4x) – cos (4x) – 1 = cos (4x) – 1;

The function value for this case is less than -1.

b) cos (4x) ≥ 0;

y = 2cos (4x) + cos (4x) – 1 = 3cos (4x) – 1;

Largest value cos (4x) – unit:

cos (4x) ≤ 1.

Multiplying the inequality by 3 and subtracting 1, we get the largest value of the function:

3cos (4x) ≤ 3;
3cos (4x) – 1 ≤ 3 – 1;
3cos (4x) – 1 ≤ 2;
y ≤ 2;
Answer. Highest function value: 2.