How to understand the sufficient condition for global optimum? From my understanding, the global optimum should be 0 instead of $\geq 0$, where does this come from?

Consider the constrained optimization problem

$$\underset{\beta \in R^p}{\min}f(\beta) \space \space \space s.t. \space\beta \in C$$

where $f is a convex objective function, and C is a convex constraint set$

when $f$ is differentiable then a sufficient condition for a vector $\beta^* \in C$ to be a global optimum is :

$$ \langle \triangledown f(\beta^*),\beta - \beta^* \rangle \geq 0 $$ for any $\beta \in C$

How to understand the sufficient condition for global optimum? From my understanding, the global optimum should be 0 instead of $\geq 0$, where does this come from?

Consider the constrained optimization problem

$$\underset{\beta \in R^p}{\min}f(\beta) \space \space \space s.t. \space\beta \in C$$

where $f$ is a convex objective function, and $C$ is a convex constraint set

when $f$ is differentiable then a sufficient condition for a vector $\beta^* \in C$ to be a global optimum is :

$$ \langle \triangledown f(\beta^*),\beta - \beta^* \rangle \geq 0 $$ for any $\beta \in C$