The maximum slope of the curve y=-x^3 + 3x^2 + 9x-27 is:

$y=-x^3 + 3x^2 + 9x-27$

function of maximise = $y'=-3x^2+6x+9=g(x)$

$g(x)=-3(x^2-2x-3)$

$g'(x)=-6x+6=0⇒ x = 1$

$g''(x)=-6⇒(g''(1)=6)<0$ ⇒ x = 1 point of maxima

$g(max)=-3(1-2-3)=12$