A boy throws a stone in the air, its height at any time is given by $h=-3t^2+12t-4\text{m}$. The maximum height (in $\text{m}$) attained by the stone is

The correct option is B $8\text{m}$
Given, height of the stone as a function of time as $h\left(t\right)=-3t^2+12t-4$
The maximum or minimum value of the function at any time $t$ is given by
$\frac{dh\left(t\right)}{dt}=0$
$\Rightarrow \frac{d(-3t^2+12t-4)}{dt}=0\Rightarrow -6t+12=0$
$\Rightarrow t=2sec$
Now, checking whether the function has maximum or minimum value at $t=2sec$
For function to be maximum, $\frac{d^{2}h\left(t\right)}{d{t}^2}<0\Rightarrow \frac{d(-6t+12)}{dt}=-6<0$
Thus, function has maximum value at $t=2sec$
Hence, maximum height of the stone is
$h\left(2\right)=-3(2{)}^2+12\times 2-4=8\text{m}$