Explanation

(a)
Given, $y = - x^{3} + 3x^{2} + 2x - 27$
$\Rightarrow \ \ \ \ \ \frac{dy}{dx} = - 3x^{2} + 6x + 2$
Let slope $z = \frac{dy}{dx} = - 3x^{2} + 6x + 2$
Then, $\frac{dz}{dx} = - 6x + 6$
For maximum or minimum put $\frac{dz}{dx} = 0\$
$\Rightarrow \ \ - 6x + 6 = 0$
$\Rightarrow \ \ \ x = 1$
Now, $\frac{d^{2}z}{dx^{2}} = - 6 < 0$, maxima
The maximum value of $z$ at $x = 1$, is given by
$z = - 3 + 6 + 2 = 5$