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Optimisation - Critical Numbers for Complex Functions.

  1. Mar 11, 2015 #1
    Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows:

    Question:
    Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula:
    V = 999.87 − 0.06426T + 0.0085043T2 − 0.0000679T3

    Find the temperature at which water has its maximum density. (Round your answer to four decimal places.)

    3. The attempt at a solution

    I have done the following steps:
    Re-write equation into scientific notation:
    V(t)=-6.79x10-5T3+8.5043x10-3T2-6.426x10-2T+999.87

    Found Derivative using Power Rule:
    V'(t)=-2.037x10-4T2+1.70086x10-2T-6.426x10-2

    I need to find critical points, where V'(t)=0 or V'(t)=und.
    There are no obvious factors and I need help to find the zero's of the derivative.
    Any help would be greatly appreciated.
     
  2. jcsd
  3. Mar 11, 2015 #2

    Dick

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    It's a quadratic equation. There is a formula to find the zeros without factoring. Remember?
     
  4. Mar 11, 2015 #3
    Oh dear... of course!... I don't know why I didn't think of that. Thank you so much haha.

    The minimum of the function was 3.966514624 degrees for anyone interested.
     
    Last edited: Mar 11, 2015
  5. Mar 11, 2015 #4

    Ray Vickson

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    You were told to find the maximum, while you claim to have found the minimum.

    Have you tested whether your point is a maximum or a minimum? Just setting ##V'(T) = 0## will not tell you this; you need to use a second-order test (involving the second derivative ##V''(T)##), or use some other types of tests.

    Also: how do you know you should set the derivative to 0 at all? Perhaps the constraints ##0 \leq T \leq 30## mess things up? You need to check that as well.
     
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