in math, to disprove something, all you need is ONE counter example, and you are allowed to choose it if you can find one. here i found infinitely many counter-examples. it is easy to satisfy the equations and inequalities, but there is no minimum for the x value as the problem claimed. since i can find arbitrary a,b, and c values that fulfill all requirements, but require a smaller x value than any that i chose before. if you look at my post, you will realize that i could choose the a,b,c first and then find the x. a,b, and c are arbitrary, so i can choose any a,b,c i want, as long as they fulfill the required conditions. (your example does not fulfill the conditions by the way).
my example doesnt fulfil a^3+b^3+c^3>=3 because chosen x is wrong... you must first find x and then for every posibility a,b,c which fulfil a+b+c=3, this a^3+b^3+c^3>=3 is true... I repeat fo every combination of a,b,c , not one I found this ONE exapmle to disprove your chosen x (-50) for example x=0, can you find a,b,c, which doesnt filful equation and inequality?