Range of cos(x)=-2: Exploring Critical Points

In summary, the equation has no solution as it would be the point where y equals positive and negative infinity. However, there are defined critical points where the equation turns around. Using calculus, the author found some critical values.
  • #1
GreenPrint
1,196
0

Homework Statement



Ok so I was trying to find the range of the equation below, commen sense and simply looking at it yields that the range is set of all reals... I however used calculus to find the critical points, I've always been told that there are none when the range is the set of all reals as it would be the point were y equals positive and negative infinity, hence these points are never reached and the equation never "turns around", but I found some, I don't see what I have done wrong there are defined critical values and I believe I found them, I'm just confused becasue that would mean what exactly? infinity in the set of all reals can be defined in the complex number system... Can somone please explain to me why I got an answer and what it means, i.e. I know why I got an answer don't understand what it means... also please see my question at the bottom of the picture... also please tell me the trute lol no lies like how people are normally told

cos(x)=-2

has no solution as that is a complete lie as there are an infinite amount of solutions so to say there isn't one is just wrong lol... so the truth only please :O

Homework Equations



see below

The Attempt at a Solution


I included my work step by step just incase you think i was suppose to get a giant explosion when solving and didn't get one, because "I wasn't suppose to get a solution", I don't see what I did wrong...
http://img844.imageshack.us/img844/7654/61037261.jpg

Uploaded with ImageShack.us
 
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  • #2


as well as the large question I had the small question on the bottom and reread it sorry

sorry my english is awsome lol

but you I was taught in like 2cnd grade that the symbol "<" with a line underneath it meant equal to or less than but microsoft word tells me that less than or equal to is the symbol is "<" with a line above it...

what is the proper notation

fixed another english mistake =O ok that should be correct english lol
 
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  • #3


well infinity does have it's limits right... it was Gallalao... English... the guy who is mostly known for... Heliocentric theory or however you spell it... galileo? Was the one who said that even though you can draw a circle and draw an "infinite" amonut of lines within the circle to it's permiter... but drew a bigger circle and extended those exact lines in the smaller circle to the permiter of the bigger circle there would be space inbetween the lines... but if you just drew the bigger circle and and drew an infinite amount of lines within the circle you could cover the whole circle and would say that there was an "infinite" amount to... hmph?
 
  • #4


lets see here if I turned all of the real numbers into complex numbers...

give me a moment here let's see
 
  • #5


I'll simplify furhter but don't know about why there's an answer :O
 
  • #8


My gut tells me this can be siiplifed further but my retarted brain is fried lol it's all in the base of i now... this is were I got stuck in my other problem on that I posted on here today...

My retarded brain can't figure it out my head says like most fractions that are in terms of the same base with different exponents can be simplied

like

5^(6)/(5^2 - 5 ^4) could be simplifed with ease all of it is in terms of the same base so how come my brain can't simplify my answer above further :O

plus what does it mean :O
 
  • #10


:O... :O... :O... hmmm set answer to infinity...
equals divide by zero... :O ?...

Uh so confused
 
  • #11


Well that would explain why any real number can be expressed as a coplex number raised to some exponent and in terms of 2/pi wouldn't it? There appears to be an overlap or something...
 
  • #12


have i answered my own question or am I just being stupid? I don't know but I still would like to simplify that answer further now that it's all in terms of i... it makes sense no if infinity in the set of real numbers was really the end points of the real number system the point were real numbers and coplex numbers meet tangent to but don't intersect each other... I don't know I think my theory might be slightly correct its something along those lines? Who knows can be completely wrong... well i sure hope somebody can help me simplify to and answer my qeustion on notation as well... um and divide by zero... that line... the line were all infinities cross of all number systems

uh so confused...
 
  • #13


i think i figuered out how to divide by zero!... hmmm remarkable...
 
  • #14


You have to find the range of a function y(x) which is defined for real numbers.
The derivative you get is correct, and it can not be zero for any real x.
If you do not have real roots for y' = 0 then the function does not have minima or maxima, it does not "turn". The function is either increasing or decreasing. Find out which is true. Now the question is, between what values does the function vary. Find the horizontal and vertical asymptotes. If you have vertical ones the function goes to infinity when x approximates such an asymptote. Find out if it is - infinity or + infinity or both.

ehild
 
  • #15


ok but is there a way to simplify when it's all in terms of i like in the problem below this
 
  • #16


i knew i was just being stupid lol but can you please help me simply a fraction like that when I got it all in terms of one base such as i thanks like in the problem below that one would be the most help as it is actually practical as it's an actual answer that is correct that i got just trying to simplify further
 
  • #17


I do not understand what you did and what you want. If you have two complex numbers you can not say that one is greater than the other.

ehild
 
  • #18
  • #19


GreenPrint said:
http://img341.imageshack.us/img341/3490/capturezg.jpg

Uploaded with ImageShack.us

um this would make sense but unlike what I've been told they actually overlap? That's waht it looks like to me
The reals are a one-dimensional subset of the complex numbers. The horizontal axis of the complex plane consists of the real numbers. You should not have a circle for the reals.
 
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  • #20


GreenPrint said:
i think i figuered out how to divide by zero!... hmmm remarkable...

Why don't you give people an opportunity to respond to your original question? By my count, you posted your question and 12 additional posts within less than 3 hours.

The title of your post is "Find Range, no need for calculus for this equation", yet you jumped right in and found the derivative of the function. ehild reports that this work is correct, but it isn't relevant to this problem, nor is any of what you did with complex numbers.

All you need to do is FIND THE RANGE of the given function. To do that, set (3x - 1)/(2x2 + x - 6) = y, and solve for x.

If the equation can be solved for x for any given (and real) y, the range of the function is all real numbers.
 
  • #21


Mark44 said:
Why don't you give people an opportunity to respond to your original question? By my count, you posted your question and 12 additional posts within less than 3 hours.

The title of your post is "Find Range, no need for calculus for this equation", yet you jumped right in and found the derivative of the function. ehild reports that this work is correct, but it isn't relevant to this problem, nor is any of what you did with complex numbers.

All you need to do is FIND THE RANGE of the given function. To do that, set (3x - 1)/(2x2 + x - 6) = y, and solve for x.

If the equation can be solved for x for any given (and real) y, the range of the function is all real numbers.

=( excuse my stupidity lol... I don't buy that there are no critical points... but fine I said ok :O
 
  • #22


Whether there are critical points might be interesting, but seems to be irrelevant to this problem, if the thread title is any indication.
 

1. What is the range of cos(x)=-2?

The range of cos(x)=-2 is an empty set, or in mathematical notation, ∅. This is because the cosine function can only output values between -1 and 1, so there is no real number that can satisfy cos(x)=-2.

2. What does it mean to explore critical points?

Exploring critical points means analyzing the behavior of a function at points where its derivative is equal to 0 or undefined. These points can indicate the maximum, minimum, or inflection points of a function, and understanding them can provide insight into the overall behavior of the function.

3. How do you find critical points?

To find critical points, you need to take the derivative of the function and set it equal to 0. Then, solve for the variable to find the x-values of the critical points. It is important to also check for any values where the derivative is undefined, as these can also be critical points.

4. Can a function have multiple critical points?

Yes, a function can have multiple critical points. This usually occurs when the function has multiple local maxima or minima, or when there are inflection points where the derivative is undefined.

5. How can exploring critical points be useful?

Exploring critical points can be useful in understanding the behavior of a function, such as identifying where it reaches maximum or minimum values. This information can be applied in various fields, such as optimization problems in economics or predicting the behavior of a physical system in physics.

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