The slope of the tangent to the curve `y=-x^(3)+3x^(2)+9x-27` is maximum when `x` equals

Answer

1

Explanation

If `m` be the slope of the tangent to the given curve, then
`m=(dy)/(dx)=-3x^(2)+6x+9`
`implies(dm)/(dx)=-6x+6,(d^(2)m)/(dx^(2))=-6`
Now, `(dm)/(dx)=0implies-6x+6=0impliesx=1`
Also then `(d^(2)m)/(dx^(2))lt0`, so at `x=1` the slope `m` will be maximum.