Find any local maxima or minima of y=3x^3 -2x +1. State the intervals over which the function is increasing or decreasing.

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The function y=3x^3 -2x +1

The extreme points of y lie at the solution of y' = 0. If the solution of y' = 0 is y = c, the maximum points lie where y'' for c is negative and the minimum lies where the value of y'' is positive.

y' = 9x^2 - 2

9x^2 - 2 = 0

=> x^2 = 2/9

=> x = `-sqrt 2/3` and `+sqrt 2/3`

y'' = 18x

The local minimum is at x = `sqrt 2/3` and the local maximum is at x = `-sqrt 2/3`

The function is increasing when y' > 0 and decreasing when y'< 0

9x^2 - 2 < 0

=> 9x^2 < 2

=> x^2 < 2/9

=> `x< sqrt 2/3` and `x > -sqrt 2/3`

9x^2 - 2 > 0

=> x^2 > 2/9

=> `x > sqrt 2/3` and x `< -sqrt 2/3`

The function is decreasing in `(-sqrt 2/3, sqrt 2/3)` and increasing in `(-oo, -sqrt 2/3)U(sqrt 2/3, oo)`

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