Find the critical numbers of the function f(x) = x^6*(x - 2)^5.

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The function f(x) = x^6*(x - 2)^5. The critical numbers of the function lie at the solution of the equation f'(x) = 0

f'(x) = 6x^5*(x - 2)^5 + x^6*5*(x - 2)^4

f'(x) = 0

=> 6x^5*(x - 2)^5 + x^6*5*(x - 2)^4 = 0

=> x^5*(x - 2)^4*(6x - 12 + 5x) = 0

=> x^5*(x - 2)^4*(11x - 12) = 0

=> x = 0, x = 2 and x = 12/11

The critical points of f(x) lie at x = 0, x = 12/11 and x = 2

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