My class in pre-calculus and I attempted to solve the following problem:
At what value of x do the maximum and minimum of y = -2 cos^2 x + 2 cos x + 1 (0 <= x < 2 pi) occur?
Because this class is not familiar with the tools of differential calculus, I suggested letting
u = cos x
and thus arrived at the quadratic function
f(u) = -2u^2 + 2u + 1 = -2(u - 1/2)^2 + 3/2,
where -1 <= u <= 1. The class then understood that maximum of f occurs at
u = cos x = 1/2.
x = pi/3, 5 pi/3.
For the minimum, I graphed f, restricting the domain to [-1,1]. The result of course is a part of a parabola and the students saw that the minimum of f is -3 and this occurs at
u = cos x = -1.
x = pi.
Calculus gives the same result, but I am not sure if the method I just described always works for quadratic functions in cos x or sin x. It appeared to me to be too simple to be correct. Is there any fundamental "rule" I violated?
Jose Nilo G. Binongo Fukuoka International School, Japan