Question

How to differentiate a function and what is an increasing function?

So I have trouble calculating the following question regarding the use of differentiation.
A curve has equation `y = 3x^3 - 6x^2 + 4x + 2`
Show that the gradient is never negative.

Answer 1

To calculate the derivative of this function you need to apply three rules:

1- The derivative is linear, so:

`dy/dx = d/dx (3x^3-6x^2+4x+2) = 3 d/dx (x^3) -6 d/dx (x^2) + 4(d/dx x) + d/dx (2)`

2- The power rule stating that:

`d/dx (x^n) = nx^(n-1)`

3- The derivative of a constant is zero.

Then:

`dy/dx = 9x^2-12x+4`

so the derivative of `y(x)` is a second order polynomial that we can factorize:

`dy/dx = (3x-2)^2`

Thus we can see that:

`dy/dx >= 0`

Answer 2

`"see explanation"`

Explanation:

`"to determine if a function f(x) is increasing/decreasing"`

`• " if "f'(x)>0" then" f(x)" is increasing"`

`• "if "f'(x)<0" then f(x) is decreasing"`

`"if y increases as x increases then f(x) is increasing"`

`"if y decreases as x increases then f(x) is decreasing"`

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

`rArrdy/dx=9x^2-12x+4=(3x-2)^2`

`AAx inRR(3x-2)^2>0`

`•color(white)(x)dy/dx=m_(color(red)"tangent")`

`rArr" gradient is never negative"`

`f(x)" is increasing"`
graph{3x^3-6x^2+4x+2 [-8.89, 8.89, -4.444, 4.445]}