Inspired by Dev Sharma

Algebra Level 4

$f( x,y) + g(x,y) \geq 6 \text{ and } f(x,y) \geq 2.$

Jean can show that $f$ and $g$ are 2 functions on 2 variables $x$ and $y$ that satisfy the conditions above.

Given only that information, are you able to deduce the largest $N$ such that $g(x,y) \geq N$ for all real values of $x$ and $y$ ?

If you think that no answer exists, enter in -1000.

Inspiration .
See also .

The answer is -1000.00.

1 solution

Spencer Whitehead
Feb 25, 2016

Let's consider functions $f(x,y)$ that satisfy $f(x,y) \ge 2$ for all $x,y$ . Many people may immediately jump to see this as meaning that $f(x,y)$ has a minimum value of 2, but we could define $f(x,y)=c,c > 2$ , which certainly satisfies the inequality, while not having a minimum of 2.

Similarly, if $f(x,y)=c$ , we can define $g(x,y)=6-c$ . We then get $c+(6-c) \ge 6 \implies 6 \ge 6$ , which is certainly true. However, $c \ge 2$ , $6-c$ can be arbitrarily small, thus we cannot choose an $N$ that $g(x,y)$ is guaranteed to be larger than for all $x,y$ .

Moderator note:

More generally, we cannot just subtract inequalities like we deal with equations.

Inspired by Dev Sharma

The answer is -1000.00.

1 solution

Moderator note:

0 pending reports