The graph of the function, f(x) = 3x^2 + x + 2, opens down/up and has a maximum/minimum value.

If the function has a ∪ shape (opens up), then it only goes so low—has a minimum value.

Any even-degree function with a positive leading coefficient will open up, so have a minimum. Your function is degree 2 (even) and has a leading coefficient of 3 (positive).

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Comment on other polynomial functions

If the leading coefficient of an even-degree function is negative, it will have an inverted-U shape (∩), so will have a maximum.

Any odd-degree function will have a generally rising (/) or generally falling (\) shape, depending on the sign of its leading coefficient. If positive, then rising; if negative, then falling.