Suppose function f(x) is continuous and differentiable at x=0, and function g(x) is continuous but non-differentiable at x=0. If the function h $\left(x\right)=e^{g\left(x\right)}{f}^2\left(x\right)g\left(x\right)$ is differentiable at x=0, then which of the following is/are CORRECT?