Question

At what point, the slope of the curve $y=-x^3+3x^2+9x-27$ is maximum? Also find the maximum slope.

Answer 1

first find slope of the equation

$\frac{dy}{dx}=-3x^2+6x+9$

for this to be maximum $f^{\prime \prime }\left(x\right)=0$

$f^{\prime \prime }\left(x\right)=-6x+6=0$

$x=1$,substitute in given equation

$y=-1^3+3+9-27$

$y=-16$

the point on the curve is $(1,-16)$