To begin with, we can simplify y using the Pythagorean identity:
y = 2 cos²(x) + 3 sin²(x) + 5
y = 2 (cos²(x) + sin²(x)) + sin²(x) + 5
y = 2 + sin²(x) + 5
y = sin²(x) + 7
Next, we can further rewrite this using the half angle identity for sine:
y = (1 - cos(2x))/2 + 7
y = 15/2 - 1/2 cos(2x)
Now, since cos(x) is bounded between -1 and 1, we have
max(y) = 15/2 - 1/2×(-1) = 15/2 + 1/2 = 8
and
min(y) = 15/2 - 1/2×1 = 15/2 - 1/2 = 7
Then the sum of the maximum and minimum is 8 + 7 = 15.
