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Math Help - Need help with this math problem

  1. #1
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    Need help with this math problem

    A spot of light on a computer screen moves in a horizontal line across the screen. At time t seconds, its distance, x mm, from the left-hand edge of the screen is given, for t>/= 0, by

    x=t^3-12t^2+kt,

    where k is a positive constant. Find the set of values of k for which x is an increasing function of t. (answer: k>/=48)

    Im not sure whats the right way to go about solving this problem. Can somebody guide me along? Thank you!
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  2. #2
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    Re: Need help with this math problem

    Are you allowed to use Calculus for this problem? If so, a function is increasing where its derivative is 0. If not, draw a graph and look at where the graph is increasing...
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  3. #3
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    Re: Need help with this math problem

    Quote Originally Posted by Prove It View Post
    Are you allowed to use Calculus for this problem? If so, a function is increasing where its derivative is 0. If not, draw a graph and look at where the graph is increasing...
    This must be a typo. Prove It surely knows better! A differentiable function is increasing where its derivative is positive.
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  4. #4
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    Re: Need help with this math problem

    Yes it's a typo, I meant to write " > 0 " hahaha
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  5. #5
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    Re: Need help with this math problem

    First of all, thanks for the help guys!

    So Given that - A differentiable function is increasing where its derivative is positive , i tried putting the (derivative of x) > 0.
    so..
    3t^3-24t+k>0
    3t^2-24t>-k
    -k<3t^2-24t
    then im stuck again..

    Am i going in the right direction?
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  6. #6
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    Re: Need help with this math problem

    First of all, I assume that first line is a typo because you've fixed it in the second. Anyway, the most direct method to solve a quadratic inequality is to complete the square.
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    Re: Need help with this math problem

    Sorry for the typo!
    To clear up:
    I took x=t^3-12t^2+kt and differentiated it to get ---> 3t^2-24t+k (corrected from 3t^3-24t+k which was incorrectly written in my last post)


    Anyway, by completing the square method, i got :
    (x-4)^2=-k/3 +16

    Up till here am i doing the right thing?
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    Re: Need help with this math problem

    Well apart from the fact that it is supposed to be $\displaystyle \begin{align*} ( x - 4 ) ^2 > -\frac{k}{3} + 16 \end{align*}$, that's correct, so now

    $\displaystyle \begin{align*} \sqrt{ (x - 4)^2 } &> \sqrt{ 16 - \frac{k}{3} } \\ |x - 4| &> \sqrt{ 16 - \frac{k}{3}} \\ x - 4 < - \sqrt{ 16 - \frac{k}{3} } \textrm{ or } x - 4 &> \sqrt{ 16 - \frac{k}{3}} \\ x < 4 - \sqrt{ 16 - \frac{k}{3} } \textrm{ or } x &> 4 + \sqrt{ 16 - \frac{k}{3}} \end{align*}$
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