Interchange of trace and infimum

Question

Let be such $f( \cdot,t): \mathbb{R}^{n \times n} \to \mathbb{R}^{n \times n}$, where $t \in \mathbb{R}$. How to show that the following is true \begin{align} \text{Trace } \left(\inf_t f(X,t) \right) \le \inf_{t} \text{Trace } \left( f(X,t) \right) \end{align}

we can assume that $f(X,t) \succeq 0$ (positive).