Question

Is `f(x)=-12x^3+17x^2+2x+2` increasing or decreasing at `x=2`?

Answer 1

decreasing

Explanation:

To test if a function is increasing or decreasing at some point.

Require to evaluate f'(x) at this point

• If f'(x) > 0 then function is increasing.

• If f'(x) < 0 then function is decreasing .

`f'(x) = - 36 x^2 + 34x + 2`

`f'(2 ) = - 36(2 )^2 + 34( 2 ) + 2 = - 144 + 68 + 2 = - 74`

Since f'(x) < 0 then function is decreasing at x = - 2
graph{-12x^3+17x^2+2x +2 [-14.23, 14.24, -7.12, 7.11]}

Answer 2

Decreasing.

Explanation:

The sign of the first derivative reveals the rate of change of a function—that is, if the function is increasing or decreasing:

• If `f'(2)<0`, then `f(x)` is decreasing at `x=2`.
• If `f'(2)>0`, then `f(x)` is increasing at `x=2`.

Find `f'(x)` through the power rule.

`f(x)=-12x^3+17x^2+2x+2`

`f'(x)=-36x^2+34x+2`

Find `f'(2)`.

`f'(2)=-36(2)^2+34(2)+2=-144+68+2=-74`

Since `f'(2)<0`, the function is decreasing at `x=2`. We can check a graph of `f(x)`:

graph{-12x^3+17x^2+2x+2 [-2, 4, -100, 100]}