You need to solve the equation `(dy)/(dx) = 0` to find the critical values of the function such that:

`(dy)/(dx) = 588x^3 + 84x^2 - 24x`

You need to solve the equation `588x^3 + 84x^2 - 24x = 0` such that:

`588x^3 + 84x^2 - 24x = 0`

`98x^3 +...

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You need to solve the equation `(dy)/(dx) = 0` to find the critical values of the function such that:

`(dy)/(dx) = 588x^3 + 84x^2 - 24x`

You need to solve the equation `588x^3 + 84x^2 - 24x = 0` such that:

`588x^3 + 84x^2 - 24x = 0`

`98x^3 + 14x^2 + 4x = 0`

`49x^3 + 7x^2 + 2x = 0`

You need to factor out x such that:

`x(49x^2 + 7x + 2) = 0`

Notice that only x = 0 since `49x^2 + 7x + 2 gt 0` for any real x.

You need to select a value for x, smaller than 0 such that:

`x = -1 =gt x(49x^2 + 7x + 2) lt 0`

You need to select a value for x, larger than 0 such that:

`x = 1 =gt x(49x^2 + 7x + 2)gt 0`

**Hence, the function reaches its absolute minimum over interval [-20,20] at x = 0.**