What is the range of the function y=sqrt(1-cosxsqrt(1-cosx(sqrt(1-cosx ......oo ?
3 Answers
I need double-check.
Explanation:
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Explanation:
Given:
y = sqrt(1-cos xsqrt(1-cos xsqrt(1-cosxsqrt(...))))
write
y = sqrt(1-tsqrt(1-tsqrt(1-tsqrt(...))))
Square both sides to get:
y^2 = 1-tsqrt(1-tsqrt(1-tsqrt(...))) = 1-ty
Add
y^2+ty-1 = 0
This quadratic in
y = (-t+-sqrt(t^2+4))/2
Note that we need to choose the
So:
y = (-t+sqrt(t^2+4))/2
Then:
(dy)/(dt) = -1/2+t/(2sqrt(t^2+4))
This is
t/sqrt(t^2+4) = 1
That is:
t = sqrt(t^2+4)
Squaring both sides:
t^2 = t^2+4
So the derivative is never
So the maximum and minimum values of
When
y = (1+sqrt(5))/2
When
y = (-1+sqrt(5))/2
So the range of
[(-1+sqrt(5))/2, (1+sqrt(5))/2]
graph{(y-(-(cos x)+sqrt((cos x)^2+4))/2) = 0 [-15, 15, -0.63, 1.87]}
See below.
Explanation:
We have
Here
Now
then the feasible limits are
NOTE
With
we have that