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Math Help - Find the critical numbers?

  1. #1
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    Exclamation Find the critical numbers?

    How to find the critical number of the given function:
    f(x)= x^3(x-2)^2

    Attempt:
     f'(x)=(x^3)(2[x-2][1])+(x-2)^2(3x^2)

    Now we have to equate f'(x) to zero

     f'(x)=(x^3)(2[x-2])+(x-2)^2(3x^2) =0

    After this step I dont know how to simplify to get the x co-ordinates or critical numbers

    Please answer in steps.

    Thank you
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  2. #2
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    <br />
f'(x)=(x^3)(2(x-2))+(x-2)^2(3x^2) =x^2\left(2x(x-2)+(x-2)^23\right)

    <br />
=x^2(x-2)\left(2x+3(x-2)\right)=x^2(x-2)(5x-6)<br />
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  3. #3
    Senior Member DeMath's Avatar
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    Quote Originally Posted by mj.alawami View Post
    How to find the critical number of the given function:
    f(x)= x^3(x-2)^2

    Attempt:
     f'(x)=(x^3)(2[x-2][1])+(x-2)^2(3x^2)

    Now we have to equate f'(x) to zero

     f'(x)=(x^3)(2[x-2])+(x-2)^2(3x^2) =0

    After this step I dont know how to simplify to get the x co-ordinates or critical numbers

    Please answer in steps.

    Thank you
    Hint:

    f\left( x \right) = {x^3}{\left( {x - 2} \right)^2} = {x^3}\left( {{x^2} - 4x + 4} \right) = {x^5} - 4{x^4} + 4{x^3}.

    f'\left( x \right) = 5{x^4} - 16{x^3} + 12{x^2} = {x^2}\left( {5{x^2} - 16x + 12} \right) = {x^2}\left( {5x - 6} \right)\left( {x - 2} \right).
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