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Math Help - Finals question help! (calculus)

  1. #1
    Newbie rIBBON:lacedx's Avatar
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    Finals question help! (calculus)

    Hi everyone,

    My Calculus teacher gave us some review sheets of what to expect on the final. I've narrowed it down to this question that I can't figure out no matter what. Can anyone help me? Here is the problem:

    Find the value of x where f(x)= x^2-3(x+2)^(1/2) has its absolute minimum.

    I know I'm supposed to take the derivative and set that equal to zero to find the critical points. However, I keep getting stuck and can't seem to find the critical points because I get lost in the math.

    Thank you for any help!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by rIBBON:lacedx View Post
    Hi everyone,

    My Calculus teacher gave us some review sheets of what to expect on the final. I've narrowed it down to this question that I can't figure out no matter what. Can anyone help me? Here is the problem:

    Find the value of x where f(x)= x^2-3(x+2)^(1/2) has its absolute minimum.

    I know I'm supposed to take the derivative and set that equal to zero to find the critical points. However, I keep getting stuck and can't seem to find the critical points because I get lost in the math.

    Thank you for any help!
    I will assume that by absolute minimum you mean global minimum.

    f(x)=x^2-3(x+2)^{1/2}

    f'(x)=2x-3(1/2)(x+2)^{-1/2}=2x-\frac{3}{2(x+2)^{1/2}}

    setting f'(x)=0 gives us:

    2x-\frac{3}{2(x+2)^{1/2}}=0\ \ \ \ \ ..(1)

    simplifying:

    \frac{4}{3}x=\frac{1}{(x+2)^{1/2}}

    or after squaring:

    \frac{16}{9}x^2 (x+2)-1=0

    Now this is a cubic, which has three roots, of these two are spurious (that
    is are not zeros of the derivative of f(x), and the other is
    a genuine root. It is probably better to go for a direct numerical solution of
    (1) rather than solve this cubic.

    A quick sketch of f(x) shows the minimum is close to x=0.5,
    and formula itteration of:

    x_{n+1}=\frac{3}{4(x_n+2)^{1/2}}

    gives the minimum occurs at x \approx 0.4766 and is \approx -4.494

    RonL
    Last edited by CaptainBlack; January 7th 2007 at 11:08 PM.
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