Prove that function f(x) =`{-2x^(3)+3x^(2)-6x+5,xlt0` `-x^(2)-x+1, xge0` is decreasing for all x.

Explanation

Since lim f(x) =5 and lim f(x) =1 , f(x) is discontinuous at x=1
Now let us check the nature of the branch functions
f'(x) = `{{:(-2x^(3) + 3x^(2) - 6x + 5 "," x lt 0) , (-x^(2) -x + 1"," " " xge 0):}`
= `{{:(-6((x-1//2)^(2) + 3//4) ","x lt 0) , (-2x-1"," " " x gt 0):}`
Clearly derivative of each branch function is negative for corresponding values of x .
Also , `underset(xto0) ("lim") f(x) lt underset(x to 0)("lim") f(x)`
So , f(x) is decreasing for all real x including x = 0 .