Question

The maximum slope of the curve y = −x³ + 3x² + 9x − 27 is _________________.

Answer 1

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

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

∴ m = $\frac{dy}{dx}=-3x^2+6x+9$

Differentiating both sides with respect to x, we get

$\frac{dm}{dx}=-6x+6$

For maxima or minima,

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

$\Rightarrow -6x+6=0$

$\Rightarrow x=1$

Now,

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

So, x = 1 is the point of local maximum.

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

∴ Maximum value of the slope

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

= −3 + 6 + 9

= 12

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


The maximum slope of the curve y = −x³ + 3x² + 9x − 27 is ___12___.