Maximum slope of the curve y = -x³ + 3x² + 9x - 27 is

Concept:

Slope m of the curve is given by dy/dx = 0

And, condition for slope to be maximum: d2y/dx2 = 0

(dy/dx)x = a gives the value of maximum slope.

Calculation:

y = – x3 + 3x2 + 9x – 27

dy/dx = – 3x2 + 6x + 9 = slope of the curve

Now, double differentiation:

d2y/dx2 = – 6x + 6 = – 6 (x – 1)

d2y/dx2 = 0

⇒ – 6 (x – 1) = 0

⇒ x = 1

clearly, d3y/dx3 = – 6 < 0 for all value of x

∴ The slope is maximum when x = 1.

(dy/dx)x = 1 = – 3 (1)2 + 6 × 1 + 9 = 12