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

##### 2 Answers

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]}

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)=-12x^3+17x^2+2x+2#

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

Find

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

Since

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