Explanation

(c)
Let $x$ and $y$ be a two parts. Then,
$x + y = 12$
Let $P = x^{2}y^{4}$
$\Rightarrow \ \ \ \ \ \sqrt{p} = xy^{2} = L$ [say]
$\Rightarrow \ \ \ \ \ \ L = x(12 - x)^{2}$
$\Rightarrow \ \ \ \ \ \frac{dL}{dx} = (12 - x)^{2} - 2x(12 - x)$
For maxima, pot $\frac{dL}{dx} = 0$
$\Rightarrow \ \ \ (12 - x)\lbrack 12 - 3x\rbrack = 0$
$\Rightarrow \ \ \ \ \ x = 12,\ x = 4$
At $x = 4$, it is maximum $(\because x \neq 12)$
Hence, it is maximum, when the parts are 4, 8