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saubhik

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  • 2 years ago
  • 2 years ago

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  1. saubhik
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    I find this rather important to some people of openstudy who claim that "anything log,exponentials,arcsin,arccos,arctan...etc can be taken on both sides of an inequality". I hereby state the general principle. Suppose we have an inequality over real x and y: x>y. The above implies f(x)>f(y) if and only if f is a strictly increasing function in [x,y]. The above implies f(x)<f(y) if and only if f is a strictly decreasing function in [x,y]. The above implies f(x)=f(y) if and only if f is a constant function in [x,y]. {For x>=y the above statements holds after striking off the "strictly" word.} Now we have the common usage "take log on both sides" whenever required in some inequalities. Observe that f(x)=log(x) is an increasing function in [0,+infty). (Remember you can't "take log on both sides" when there is a negative quantity in atleast one side---see the domain of function definition.) However "take exponential on both sides" is always allowed since the function f(x)=e^x is increasing everywhere {i.e. in (-infty,+infty)}. Even the smallest things you apply to inequalities, like adding one on both sides or subtracting 1 on both sides are possible because the functions f(x)=x+1 and g(x)=x-1 are everywhere increasing. "Taking square roots on both sides" is permissible when both sides have non-negative numbers. Also the sign of the original inequality is preserved since f(x)=sqrt(x) is increasing in [0,+infty). While "taking trigonometric functions" be cautious about the intervals where the function is increasing or decreasing. For x,y in [-pi/2,pi/2] x>y implies sinx>siny and for x,y in [pi/2,3pi/2] x>y implies sinx<siny. (You know why, i guess.) The gist of this note was to say that when operating using inequalities you are actually taking functions---you must be careful about the function's domain of definition and also the behaviour of the function in that specified interval. Of course, some operations include using functions on only one side of inequality (this is what differentiates equalities from inequalities) like x>y implies x>y-1 {x,y are reals}. You also need precautions when doing this i.e. x>y implies f(x)>y only when f(x) has an maps x to a "greater" number say x1 so that x1>x>y i.e. f(z)>z for all z in neighbourhood of x {observe that this operation dilates the inequality; using an operation to make an inequality more stronger than given requires handling with utmost care so that the inequality does not change its sign!}. P.S.:I am sorry as this was not a question as you were expecting, but you may give your suggestions. Since this was not a question, I will give a medal to everyone who comments for about 1 day, after which I will delete this message, unless you persist. Thank you for reading. Good luck!

    • 2 years ago
  2. saubhik
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    attchment for better reading..

    • 2 years ago
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  3. Muallimah
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    lol

    • 2 years ago
  4. saubhik
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    medals for evryone :)

    • 2 years ago
  5. Muallimah
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    :p

    • 2 years ago
  6. some1
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    good piece of writing. expand more starting from the line: maps x to a "greater number". like explain it more.

    • 2 years ago
  7. saubhik
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    where r u stuck some1?

    • 2 years ago
  8. a_kaffash
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    Haaa?

    • 2 years ago
  9. saubhik
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    1 comment = 1 medal ! comment evryone.... :)

    • 2 years ago
  10. Lovelesstime
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    OMG SO LONG Medal now.

    • 2 years ago
  11. chaguanas
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    This speech is longer than Fidel Castro's marathon rants.

    • 2 years ago
  12. czaee
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    what on earth was that?? i didnt understand even a bit :P

    • 2 years ago
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