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Math Help - Minimum value of this particular trigonomterical function : really interesting

  1. #1
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    Thumbs up Minimum value of this particular trigonomterical function : really interesting

    First take a look at the problem :

    Find the minimum value of the trigonometrical function : y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2}

    If we proceed by perfect-square method , we get ,

    {(2\sin x - 3/ \ sin x)}^{ 2} \geqslant 0

    => {(2\sin x - 3/ \ sin x)}^{ 2} + 12 \geqslant 12

    => {(2\sin x - 3/ \ sin x)}^{ 2} + 2  \cdot  2\sin x \cdot 3 (1/ \sin x)   \geqslant 12

    Therefore , y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2} \geqslant 12

    But this minimum value will be achieved if and only if ,
    {(2\sin x - 3/ \ sin x)}^{ 2} = 0

    which when solved gives   \sin x = 1.2247  ( approximately ) , which is certainly not possible since maximum value of \sin x  is 1

    Does this mean that the function y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2} does not have any minimum value .

    But if we plot the graph of y = 4{\sin }^{2 }x + 9{\(1/sin x) }^{2} , it looks like it has a minimum value of 13 .

    Please analyze and explain !
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  2. #2
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    The function 4x + 9/x decreases when 0 < x <= 1. Therefore, 4sin^2(x) + 9/sin^2(x) has a minimum when sin^2(x) = 1, and this minimum is indeed equal to 13.

    To prove that 4x + 9/x decreases, you can take 0 < x1 < x2 <= 1 and compare 9/x1 - 9/x2 and 4(x2 - x1).
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  3. #3
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    Your explanation is not quite clear . Moreover , if 4x + 9/x decreases when 0 < x <= 1 , how does it imply that 4sin^2(x) + 9/sin^2(x) has a minimum when sin^2(x) = 1 ? You'll have to state the reason if any . And further more , you've found out the min value of 4sin^2(x) + 9/sin^2(x) which is not exactly what i posted . Please read the question carefully . Graph of sin and cosec are distinctly different!!
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  4. #4
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    if 4x + 9/x decreases when 0 < x <= 1 , how does it imply that 4sin^2(x) + 9/sin^2(x) has a minimum when sin^2(x) = 1 ?
    The function sin^2(x) assumes all values between 0 and 1 and only them. Therefore, the minimum of 4sin^2(x) + 9/sin^2(x) on all real line equals the minimum of 4x + 9/x when 0 < x <= 1. Moreover, the minimum of 4x + 9/x on this interval is assumed when x = 1, so the minimum of 4sin^2(x) + 9/sin^2(x) is assumed when sin^2(x) = 1. If you need more explanation, please say if this is not intuitively clear or if you are looking for formal reasons, such as how this can be deduced from the axioms of real numbers. (Deduction from axioms, even though it is the most air-tight proof, is often not the clearest.)

    And further more , you've found out the min value of 4sin^2(x) + 9/sin^2(x) which is not exactly what i posted .
    What is the difference?
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  5. #5
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    emakarov did state the reason! He said that 4x+ 9/x is decreasing for 0< x< 1 and suggested how you could see that for yourself. Since 4(1)+ 9/(1= 13, the minimum value of 4x+ 9/x, or any 4f(x)+ 9/f(x) where f(x) lies between 0 and 1.
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  6. #6
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    Quote Originally Posted by emakarov View Post
    The function sin^2(x) assumes all values between 0 and 1 and only them. Therefore, the minimum of 4sin^2(x) + 9/sin^2(x) on all real line equals the minimum of 4x + 9/x when 0 < x <= 1. Moreover, the minimum of 4x + 9/x on this interval is assumed when x = 1, so the minimum of 4sin^2(x) + 9/sin^2(x) is assumed when sin^2(x) = 1. If you need more explanation, please say if this is not intuitively clear or if you are looking for formal reasons, such as how this can be deduced from the axioms of real numbers. (Deduction from axioms, even though it is the most air-tight proof, is often not the clearest.)

    What is the difference?
    Beautiful explanation ! Totally clear now .

    But are you aware ( please let me know! ) why such a , what should I say , Paradox is happening when we use the perfect square method ?
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  7. #7
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    You correctly (and elegantly) determined that when x > 0, f(x) = 4x^2 - 9/x^2 >= 12 and f(x) = 12 iff x^2 = 3/2 (i.e., x is about 1.225). However, g(x) = sin^2(x) is not able to reach 1.5; it can only reach 1. Since f(x) decreases when 0 < x <= 1, having g(x) = 1 would provide you with the minimum value for the composition f(g(x)) = 4sin^2(x) + 9/sin^2(x).

    The perfect square method worked correctly for the function f(x). However, it knows nothing about the restriction 0 <= g(x) <= 1, so you can't expect it to take this restriction into account.
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