Question

Maximum slope of the curve y = -x³ + 3x² + 9x - 27 is
(a) 0 (b) 12 (c) 16 (d) 32

Answer 1

The given curve is y = −x³ + 3x² + 9x − 27.

Slope of the curve, m = $\frac{dy}{dx}$

∴ m = $\frac{dy}{dx}=-3x^2+6x+9$

⇒ $\frac{dm}{dx}=-6x+6$

For maxima or minima,

$\frac{dm}{dx}=0$

$\Rightarrow -6x+6=0$

$\Rightarrow x=1$

Now,

$\frac{d^{2}m}{d{x}^2}=-6$ < 0

⇒ x = 1 is the point of local maximum

So, the slope of the given curve is maximum when x = 1.

∴ Maximum value of the slope, m

$=-3\times {\left(1\right)}^2+6\times 1+9$ (m = −3x² + 6x + 9)

$=-3+6+9$

$=12$

Thus, the maximum slope of the curve y = −x³ + 3x² + 9x − 27 is 12.

Hence, the correct answer is option (b).