jmc

For $x\le -5,$ $$|x+2|+|x+4|+|x+5|=-(x+2)-(x+4)-(x+5)=-3x-11.$$For $-5\le x\le -4,$ $$|x+2|+|x+4|+|x+5|=-(x+2)-(x+4)+(x+5)=-x-1.$$For $-4\le x\le -2,$ $$|x+2|+|x+4|+|x+5|=-(x+2)+(x+4)+(x+5)=x+7.$$For $x\ge -2,$ $$|x+2|+|x+4|+|x+5|=(x+2)+(x+4)+(x+5)=3x+11.$$Thus, the function is decreasing on $(-\infty ,4]$ and increasing on $[4,\infty ),$ so the minimum value occurs at $x=-4,$ which is $\boxed{3}.$