The polynomial function is

#f(x)=-2x^3+6x^2+18x-18#

This function is defined, continuous and derivable on #RR#

The first derivative is

#f'(x)=-6x^2+12x+18#

The critical points are when #f'(x)=0#

#-6(x^2-2x-3)=0#

#(x+1)(x-3)=0#

#f'(x)=-(x+1)(x-3)#

Therefore,

#x=-1# and #x=3#

We can make a variation chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaa)##3##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x+1##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##x-3##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##f'(x)##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaa)##↘##color(white)(aaaa)##↗##color(white)(aaa)##↘#

The second derivative is

#f''(x)=-12x+12#

#f''(-1)=12+12=24#, #=>#, this is a relative minimum as #f''(-1)>0#

#f''(3)=-12*3+12=-24#, #=>#, this is a relative maximum as #f''(-1)<0#

graph{-2x^3+6x^2+18x-18 [-74.1, 74.1, -37.03, 37]}