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Topic: max/min of quadratic trigonometric functions
Replies: 3   Last Post: Feb 11, 1998 6:18 AM

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J. N. G. Binongo

Posts: 16
Registered: 12/6/04
max/min of quadratic trigonometric functions
Posted: Feb 9, 1998 9:50 AM
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My class in pre-calculus and I attempted to solve the following

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.

That is,

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.

That is,

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

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