# Is #f(x)=(-2x^3-x^2-5x+2)/(x+1)# increasing or decreasing at #x=0#?

##### 1 Answer

decreasing at x = 0

#### Explanation:

To test if a function is increasing / decreasing at x = a , require to check the sign of f'(a)

• If f'(a) > 0 then f(x) is increasing at x = a

• If f'(a) < 0 then f(x) is decreasing at x = a

Require to find f'(x)

differentiate using the

#color(blue)" Quotient rule " # If

# f(x) = g(x)/(h(x)) " then " f'(x) = (h(x).g'(x) - g(x).h'(x))/(h(x))^2 #

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

#g(x) = -2x^3 - x^2- 5x + 2 rArr g'(x) =-6x^2-2x-5# h(x) = x+1

# rArr h'(x) = 1#

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

substitute these values into f'(x)

#rArr f'(x) =( (x+1).(-6x^2-2x-5) - (-2x^3-x^2-5x+2).1)/(x+1)^2# and f'(0) =

#(1.(-5) - 2.1)/1 = -7 # since f'(0) < 0 , f(x) is decreasing at x = 0

Here is the graph of f(x)

graph{(-2x^3-x^2-5x+2)/(x+1) [-10, 10, -5, 5]}