A curve has equation the follwing equation.

y = 3x^3− 6x^2+4x+2.

Show that the gradient of the curve is never negative

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`y = 3x^3-6x^2+4x+2`

The gradient of a curve is given by the its first derivative as follows.

`y' = 9x^2-12x+14`

`9x^2-12x+4`

`=9[x^2-(12/9)x+4/9]`

`=9[(x-12/18)^2 + 4/9 – (12/18)^2]` By completing square

`=9[(x-2/3)^2+4/9-4/9]`

`=9(x-2/3)^2`

We know that `(x-2/3)^2 >= 0` always for every x.

*There for y' or gradient will be greater than or equal to 0.That means gradient never become negative.*

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