Find the critical points of the function y = 2x ^ 3-6x.

y = 2x ^ 3 – 6x.

Find the derivative of the function:

y ‘= (2x ^ 3 – 6x)’ = 2 * 3x ^ 2 – 6 = 6x ^ 2 – 6.

Critical points are the points at which the derivative is zero:

y ‘= 0,

6x ^ 2 – 6 = 0,

6 (x ^ 2 – 1) = 0,

6 (x – 1) (x + 1) = 0,

x1 = -1,

x2 = 1.

When x <-1, y ‘(x)> 0, then the function is increasing.

When -1 <x <1, y ‘(x) <0, then the function is decreasing.

For x> 1, y ‘(x)> 0, then the function is increasing.

Point x = -1 is the maximum point, y (-1) = 2 * (-1) ^ 3 – 6 * (-1) = -2 + 6 = 4.

Point x = 1 is the minimum point, y (1) = 2 * 1 ^ 3 – 6 * 1 = 2 – 6 = -4.

Critical points: (-1; 4) and (1; -4).

Answer: (-1; 4) and (1; -4).