f() = 16 cos() 8 sin^2()

1. The given function f() = 16 cos() 8 sin^2() can be simplified using trigonometric identities. Since sin^2() = 1 - cos^2(), we can substitute this into the function:

f() = 16 cos() 8 (1 - cos^2())

2. Now, distribute the 8 to both terms inside the parentheses:

f() = 16 cos() 8 - 8 cos^2()

3. Rearrange the terms to get the function in standard form:

f() = -8 cos^2() + 16 cos()

4. This function represents a quadratic equation in terms of cos(). To find the maximum or minimum value of the function, we can use the vertex formula for a quadratic function in the form f(x) = ax^2 + bx + c, where the vertex is at x = -b/2a. In this case, a = -8 and b = 16:

cos() = -16 / (2 * -8) = 1

5. Therefore, the maximum or minimum value of the function occurs at cos() = 1. To find the corresponding value of f(), substitute cos() = 1 back into the function:

f() = -8(1)^2 + 16(1) = -8 + 16 = 8