Increasing Variables
It is given that , where .
For which variable, would adding 1 to the value of it, increase by the greatest amount?
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It's\quad pretty\quad obvious\quad that\quad b,d,\quad and\quad e\quad are\quad wrong\quad choices,\\ \\ so\quad it's\quad only\quad a\quad choice\quad between\quad a\quad and\quad c.\\ \\ Adding\quad 1\quad to\quad a\quad adds\quad \frac { 1 }{ b } \quad to\quad S,\\ \\ while\quad adding\quad 1\quad to\quad c\quad adds\quad \frac { 1 }{ d } \quad to\quad S.\\ \\ since\quad b\quad <\quad d,\quad if\quad we\quad get\quad the\quad cross\quad product\\ \\ we\quad find\quad out\quad that\quad \frac { 1 }{ b } \quad >\quad \frac { 1 }{ d } ,\\ \\ therefore\quad adding\quad to\quad a\quad would\quad give\quad the\quad largest\quad increase\quad in\quad S.