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

1 Answer
Apr 11, 2018

#f(x)=(x-2)(x+5)(x+2)# is decreasing at #x=-3#

Explanation:

We are given #f(x)=(x-2)(x+5)(x+2)#

Now the function is increasing at a given point, if #(df)/(dx)>0# at that point and is decreasing if #(df)/(dx)<0#.

Here #f(x)=(x-2)(x+5)(x+2)=(x+5)(x^2-4)=x^3+5x^2-4x-20#

and #(df)/(dx)=3x^2+10x-4#

at #x=-3#, we have #((df)/(dx))_(x=-3)=3(-3)^2+10(-3)-4#

= #27-30-4=-7#

Hence #f(x)=(x-2)(x+5)(x+2)# is decreasing at #x=-3#

(see graph not drawn to scale - shrunk alon #y#-axis)
graph{(x-2)(x+5)(x+2) [-10, 10, -40, 40]}