Linearizing maximization problem around $t=0$

Question

I have a maximization problem $\max f(x;t)$ given $g_i(x;t)=0$ where $t$ is a parameter (time)(exogenous). Also I know the solution to the problem for $t=0$. Can I in any way linearize (taylor expansion) near $t=0$ and use Lagrange Multipliers method to solve for $\frac{dx}{dt}$ because it will be a linear system of equations? In the actual problem $x$ is a vector but you can think of it as scalar in order to answer this question. When I naively linearize all equations around $t=0$ ($\max \frac{\partial f}{\partial x} \dot{x}+\frac{\partial f}{\partial t}$ subject to $ \frac{\partial g_i}{\partial x} \dot{x}+\frac{\partial g_i}{\partial t}=0$),the problem gets trivial and nothing can be solved.