The function of
is parabola, since the coefficient of
is positive then the graph of C is minimum at its vertex.
BUT since the problem is realistic
and asks for the minimum value of C, then minimum value of x, the number of manufactured bottle should be at minimum.
And since the fewest possible bottles to manufacture is 0 then we plug x=0 into the function...just comment if the question was arbitrary, because then the question would ask for the roots of the polynomial which are imignary.....the answer given above is assuming the answer would be realistic because for example, you cannot produce bottles....