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

Jan 23, 2016

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 .

$f ' \left(x\right) = - 36 {x}^{2} + 34 x + 2$

$f ' \left(2\right) = - 36 {\left(2\right)}^{2} + 34 \left(2\right) + 2 = - 144 + 68 + 2 = - 74$

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]}

Jan 23, 2016

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 $f ' \left(2\right) < 0$, then $f \left(x\right)$ is decreasing at $x = 2$.
• If $f ' \left(2\right) > 0$, then $f \left(x\right)$ is increasing at $x = 2$.

Find $f ' \left(x\right)$ through the power rule.

$f \left(x\right) = - 12 {x}^{3} + 17 {x}^{2} + 2 x + 2$

$f ' \left(x\right) = - 36 {x}^{2} + 34 x + 2$

Find $f ' \left(2\right)$.

$f ' \left(2\right) = - 36 {\left(2\right)}^{2} + 34 \left(2\right) + 2 = - 144 + 68 + 2 = - 74$

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

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