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Math Help - maximium value of x

  1. #1
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    maximium value of x

    Hi, I have a function for expected utility (EU) below:

    EU=0.95(1000000-40000x)^(0.5) + 0.05(250000+750000x-40000x)^(0.5)

    I need to solve for the value of x which maximises EU, I have differentiated with respect to x and set equal to zero but keep getting the wrong answer

    can anyone help me with this? the right answer is 0.836, i don't know why im finding this difficult as it should be simple.

    Thanks

    Mark
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  2. #2
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    Is it intentional that 750000x-40000x is not simplified to 710000x in the second square root?
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  3. #3
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    it should be simplified, it comes from the earlier part of the question that i had it in two parts, sorry
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  4. #4
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    We can factor out 100 (i.e., 10000 under the square root) since it does not change the maximum point. So let f(x)=0.95\sqrt{100-4x}+0.05\sqrt{25+71x}. Then f'(x)=-\frac{4\cdot0.95}{2\sqrt{100-4x}}+\frac{0.05\cdot71}{\sqrt{25+71x}}. From here, f'(x)=0 iff 3.55\sqrt{100-4x}=3.8\sqrt{25+71x}. Taking square of both sides and adding like terms, we get 1075.65x=899.25, i.e., x\approx0.836. One must also check that both roots exist and are not zero for this x because by eliminating the denominators and taking squares we may have added spurious roots.

    You can use WolframAlpha to solve this equation like this. While it is not recommended to use it to avoid solving things by hand, it can be used to verify that each of your manual steps.
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