Suppose we have a function $$g(x)=[f(x)]^n.$$ Further assume that the function has a unique critical point (maxima or minima). If we want to get the cirtical value of $x$ then we have to differentiate $g(x)$ and equate it to zero. But for function of the above form it could also mean that we put $f(x)=0$ since $g'(x)=n(f(x))^{n-1}f'(x)=0.$ So can we conclude that the minimum point of $g(x)$ will be for a value of $x$ for which $f(x)=0$. If this is wrong reasoning then please guide me where I am wrong. I will be very thankful to you.
One more thing. Since $g(x)$ is equal to multiple products of same function hence can we conclude that $g(x)$ will achieve minimum (or maximum) value when $f(x)$ achieves minimum (or maximum) value. Thank you.
There is some merit to your reasoning, but it is not the full picture. By the chain rule you correctly have
$$g'(x) = n (f(x))^{n-1} f'(x) = 0.$$
Now, for $g'(x)$ to equal zero, it suffices for $f(x)$ to be equal to zero, $\textbf{or}$ it suffices for $f'(x) = 0$. In other words, the critical points of $g(x)$ occur at the zeros of $f(x)$ and the critical points of $f(x)$.