Concept: Slope of a curve y=f(x) at a point is the value of its first derivative at that point. i.e. m=f(x) Calculation: The slope of the curve y=−x3+3x2+2x−27 will be given by: m=y′=
d
dx
(−x3+3x2+2x−27)=−3x2+6x+2 In order to maximize the slope, we must have m′=0 and m′′<0. Now, m′=0. ⇒
d
dx
(−3x2+6x+2)=−6x+6=0 ⇒x=1 And m′′=−6<0. Therefore, the slope is maximum at x=1.