- #1
Amad27
- 412
- 1
Hi,
Let y= |sin(x) + cos(x) + tan(x) + sec(x) + csc(x) + cot(x)|
Find the minimum value of "y" for all real numbers.
Graphing is not allowed, no devices, calculators whatsoever.
Its VERY hard to find where this function = 0 analytically so it is better to take two different approaches.
(1) Assume y > 0 for (-∞, ∞)
(2) Assume y <0 for (-∞, ∞)
Case 1
-----------
y' = cos(x) - sin(x) + sec^2(x) + sec(x)tan(x) - csc(x)cot(x) - csc^2(x)
The obvious step is set y' = 0, that is VERY hard to do as well.
Let u= sin(x)
Let r = sec(x)
Let o = tan(x)
Let h = cot(x)
Let g = csc(x), this y' is the same as
y' = u' - u + r^2 + r' + g' + h'
0 = u' - u + r^2 + r' + g' + h'
u = u' + r^2 + r' + g' + h' [y' = 0 when this takes place]
This is kind of impossible to figure out. So Ill try this,
y' = cos(x) - sin(x) + sec^2(x) + sec(x)tan(x) - csc(x)cot(x) - csc^2(x)
0 = cos(x) - sin(x) + sec^2(x) + sec(x)tan(x) - csc(x)cot(x) - csc^2(x)
0 = cos(x) - sin(x) + sec(x)[sec(x) + tan(x)] - csc(x)[csc(x) + cot(x)]
0 = cos(x) - sin(x) + [sec(x) + tan(x)]/cos(x) - [csc(x) + cot(x)]/sin(x)
0 = cos^2(x) - cos(x)sin(x) + [sec(x) + tan(x)] - cos(x)[csc(x) + cot(x)]/sin(x)
0 = sin(x)cos^2(x) - cos(x) + sin(x)[sec(x) + tan(x)] - cos(x)[csc(x) + cot(x)]
[sec(x) + tan(x)] = [1 + sin(x)]/[cos(x)]
0 = sin(x)cos^2(x) - cos(x) + sin(x)[1 + sin(x)]/[cos(x)] - cos(x)[1+ cos(x)]/[sin(x)]
Thats all I can do for now.
Let y= |sin(x) + cos(x) + tan(x) + sec(x) + csc(x) + cot(x)|
Find the minimum value of "y" for all real numbers.
Graphing is not allowed, no devices, calculators whatsoever.
Its VERY hard to find where this function = 0 analytically so it is better to take two different approaches.
(1) Assume y > 0 for (-∞, ∞)
(2) Assume y <0 for (-∞, ∞)
Case 1
-----------
y' = cos(x) - sin(x) + sec^2(x) + sec(x)tan(x) - csc(x)cot(x) - csc^2(x)
The obvious step is set y' = 0, that is VERY hard to do as well.
Let u= sin(x)
Let r = sec(x)
Let o = tan(x)
Let h = cot(x)
Let g = csc(x), this y' is the same as
y' = u' - u + r^2 + r' + g' + h'
0 = u' - u + r^2 + r' + g' + h'
u = u' + r^2 + r' + g' + h' [y' = 0 when this takes place]
This is kind of impossible to figure out. So Ill try this,
y' = cos(x) - sin(x) + sec^2(x) + sec(x)tan(x) - csc(x)cot(x) - csc^2(x)
0 = cos(x) - sin(x) + sec^2(x) + sec(x)tan(x) - csc(x)cot(x) - csc^2(x)
0 = cos(x) - sin(x) + sec(x)[sec(x) + tan(x)] - csc(x)[csc(x) + cot(x)]
0 = cos(x) - sin(x) + [sec(x) + tan(x)]/cos(x) - [csc(x) + cot(x)]/sin(x)
0 = cos^2(x) - cos(x)sin(x) + [sec(x) + tan(x)] - cos(x)[csc(x) + cot(x)]/sin(x)
0 = sin(x)cos^2(x) - cos(x) + sin(x)[sec(x) + tan(x)] - cos(x)[csc(x) + cot(x)]
[sec(x) + tan(x)] = [1 + sin(x)]/[cos(x)]
0 = sin(x)cos^2(x) - cos(x) + sin(x)[1 + sin(x)]/[cos(x)] - cos(x)[1+ cos(x)]/[sin(x)]
Thats all I can do for now.