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Math Help - The greatest value

  1. #1
    Super Member dhiab's Avatar
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    The greatest value

    Find the greatest value of E :
     <br />
E=xy+x\sqrt{y^{2}-1}+y\sqrt{1-x^{2}}-\sqrt{(1-x^{2})(1-y^{2})}<br />
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  2. #2
    Member
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    First, identify any constraints on the variables. For example, any expression inside a square root has to be greater than or equal to 0.

    y^2 - 1 \geq 0
    y^2 \geq 1
    y \leq -1 or y \geq 1

    1 - x^2 \geq 0
    -x^2 \geq -1
    x^2 \leq 1
    -1 \leq x \leq 1

    (1 - x^2)(1 - y^2) \geq 0
    -1 \leq x \leq 1 and -1 \leq y \leq 1, or ( x \leq -1 or x \geq 1) and ( y \leq -1 or y \geq 1)

    Let's examine -1 \leq y \leq 1. We know y \leq -1 or y \geq 1. Therefore, -1 < y < 1 is false so the only solutions are y = -1 or y = 1.

    Now let's examine x \leq -1 or x \geq 1. We know -1 \leq x \leq 1. Therefore, x < -1 and x > 1 are false so the only solutions are x = -1 or x = 1.

    Now we can replace -1 \leq y \leq 1 with y = -1 or y = 1 and x \leq -1 or x \geq 1 with x = -1 or x = 1 to produce:

    -1 \leq x \leq 1 and ( y = -1 or y = 1), or ( x = -1 or x = 1) and ( y \leq -1 or y \geq 1)

    From here, we can replace y with -1 and determine for which value of x does the simplified expression have a maximum. Then, we can repeat the replace y with 1 and repeat the procedure. Finally, we can compare those results to replacing x with -1 and 1 and finding for which value of y does this simplified expression have a maximum.

    If you're allowed to use calculus, then I know I much more simple method.
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