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Math Help - Finding the critical numbers

  1. #1
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    Finding the critical numbers

    Find critical numbers for x^(4/5) * (x-1)^2?

    If you could help out, please. I've already done most of the work, but I'm not too sure on where to go and how to work with the problem to get the critical numbers.

    (1). The derivative of the problem. Using the product rule you get

    x^(4/5) * 2(x-1) + (x-1)^2 * 4/5x^-1/5

    here's where I'm stuck.. the factoring

    Relating to the solutions manual for a similar problem ( x^(4/5) * (x-4)^2 ), I know that I'll be able to factor it by:

    1/5x^-1/5 * (x-1)

    This is where I'm stuck... factoring will I get:

    (1/5)x^-(1/5)*(x-1) [ 5 * x * 1 + (x-1) *4] ????

    can anyone give me a push on this? I should be able to solve this step after the factoring.

    -Thanks
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  2. #2
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    Hello, endiv!

    There's some tricky algebra in this one . . .


    Find critical numbers for: . f(x) \:=\:x^{\frac{4}{5}}(x-1)^2
    Differentiate and equal to zero . . .

    . . f\,'(x) \;=\;x^{\frac{4}{5}}\cdot2(x-1) + \tfrac{4}{5}x^{-\frac{1}{5}}(x-1)^2 \;=\;0

    We have: . 2x^{\frac{4}{5}}(x-1) + \frac{4(x-1)^2}{5x^{\frac{1}{5}}} \;=\;0

    Multiply by 5x^{\frac{1}{5}}\!:\;\;10x(x-1) + 4(x-1)^2 \;=\;0

    Factor: . 2(x-1)\bigg[5x+2(x-1)\bigg] \;=\;0 \quad\Rightarrow\quad 2(x-1)(7x-2) \;=\;0


    Therefore, the critical values are at: . x \;=\;1,\:\tfrac{2}{7}

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, endiv!

    There's some tricky algebra in this one . . .


    Differentiate and equal to zero . . .

    . . f\,'(x) \;=\;x^{\frac{4}{5}}\cdot2(x-1) + \tfrac{4}{5}x^{-\frac{1}{5}}(x-1)^2 \;=\;0

    We have: . 2x^{\frac{4}{5}}(x-1) + \frac{4(x-1)^2}{5x^{\frac{1}{5}}} \;=\;0

    Multiply by 5x^{\frac{1}{5}}\!:\;\;10x(x-1) + 4(x-1)^2 \;=\;0

    Factor: . 2(x-1)\bigg[5x+2(x-1)\bigg] \;=\;0 \quad\Rightarrow\quad 2(x-1)(7x-2) \;=\;0


    Therefore, the critical values are at: . x \;=\;1,\:\tfrac{2}{7}


    holy.. wow... I didn't think of it that way! You even did the extra of giving the critical numbers. (embarassed)

    Thanks so much!
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