The maximum slope of the curve `y=-x^3+3x^2+9x-27` is 0 (b) 12 (c) 16 (d) 32

Options

A. 0
B. 12
C. 16
D. 32

Answer

B

Explanation

Let `f(x)=-x^(3)+3x^(2)+9x-27`
The slope of this curve `f'(x)=-3x^(2)+6x+9`
Let `g(x)=f'(x)=-3x^(2)+6x+9`
On differentiating both sides w.r.t `x` we get
`g'(x)=-6x+6`
For maxima or minima put `g'(x)=0impliesx=1`
Now `g''(x)=-6lt0` and hence at `x=1, g(x)` (slope) will have maximum value.
`:.` Maximum slope `=-3xx1+6(1)+9=12`