# Is #f(x)=(x^3+3x^2-4x-9)/(x+1)# increasing or decreasing at #x=0#?

##### 1 Answer

It's increasing, because the slope is positive at

graph{(x^3 + 3x^2 - 4x - 9)/(x+1) [-5, 5, -12, 5]}

The first derivative tells you whether the function is decreasing (

For this, I would actually try to do some quick synthetic division to simplify it if possible. I would guess

#ul(-1)|" "1" "3" "-4" "-9#

#"-----------------------------------"#

#" "" "" "1#

Multiply the divisor (the factor you guessed,

#ul(-1)|" "1" "3" "-4" "-9#

#" "" "" "" "-1#

#"-----------------------------------"#

#" "" "" "1#

Add the current column, and repeat the previous steps:

#ul(-1)|" "1" "3" "-4" "-9#

#" "" "" "" "-1#

#"-----------------------------------"#

#" "" "" "1" "2#

#ul(-1)|" "1" "3" "-4" "-9#

#" "" "" "" "-1" "-2#

#"-----------------------------------"#

#" "" "" "1" "2" "-6#

#ul(-1)|" "1" "3" "-4" "-9#

#" "" "" "" "-1" "-2" "" "6#

#"-----------------------------------"#

#" "" "" "1" "2" "-6" "-3#

You started with a cubic, so you end up with a quadratic. The remainder is

#x^2 + 2x - 6 - 3/(x+1)#

This is indeed easier to differentiate, so now:

#d/(dx)[x^2 + 2x - 6 - 3/(x+1)]#

#= 2x + 2 + 3/(x+1)^2#

Now, plug in

#f'(0) = |[(dy)/(dx)]|_(x=0) = 2(0) + 2 + 3/(0+1)^2 = 5 > 0#

Therefore, the function is **increasing** at