number of subsegments that contain A[i] as the minimum
You haven't stated what a subsegment is, but I assume that you mean a "contiguous subsequence", that is, $A[s],A[s+1],\ldots,A[t]$. Also, I'm assuming that $A[i]$ is still a minimum if there are other equal values (you can easily adapt the algorithm for the other option).

Let $\ell \leq i$ be the smallest index such that $A[s] \leq A[i]$ for all $\ell \leq s \leq i$, and let $r \geq i$ be the largest index such that $A[r] \leq A[i]$ for all $i \leq s \leq r$. Then $A[i]$ is a minimum of $A[s],\ldots,A[t]$ iff $\ell \leq s$ and $t \leq r$. Thus there are $i-\ell+1$ options for $s$, namely $\ell,\ell+1,\ldots,i$, and similarly $r-i+1$ options for $t$, for a total of $(i-\ell+1)(r-i+1)$.

You can easily find $\ell,r$ in linear time by scanning left of and right of $i$ (respectively), so this algorithm runs in linear time.