For minimum,
equating
$2 + x2 = 6 - 3x$
$x2 + 3x - 4 = 0$
$x = 1, -4$
Since $x > 0,$ so value occurs at $x = 1.$
At $x = 1$
$2+x^{2}=3$
$6 - 3x = 3.$
it means the largest value of the function $min( 2 + x^2 , 6 − 3x)$
$min( 3, 3)$ is $3$
The correct option is C.