• anonymous
f(x)=(x^2 -4)^ 2/3 critical numbers and point of inflection
  • Stacey Warren - Expert
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  • chestercat
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  • anonymous
For the critical points: Taking the derivative, you get f'(x) = (2/3)*(x^2 - 4)^(-1/3)*2x. Setting this equal to 0 and dividing out the (x^2 - 4)^(-1/3), you will see that x = 0. For the point of inflection: Take the derivative again, either by product or quotient rule and you'll get: (12(x^2-4)^(1/3) - 8*x^2*(x^2-4)^(-2/3))/(9*(x^2-4)^(2/3))=0. Get rid of the bottom by multiplying it to the other side then divide by (x^2-4)^(1/3) to get: 12-(8x^2)/(x^2-4) = 0. You can simplify further to eventually get x^2 = 12. Therefore the points of inflection are x = +-sqrt(12)

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