Why so serious 1

Rearranging $3^x+4^x+12^x - 16^x-9^x = 1$ , we get $3^x+4^x+12^x = 16^x+9^x+1$ , where each terms of both the LHS and RHS is $>0$ . Therefore, we can apply AM-GM inequality.

• LHS $\quad 3^x+4^x+12^x \ge 3\sqrt [3] {144^x} = 3$ as equality occurs when $3^x=4^x=12^x=1$ or $x=0$
• RHS: $\quad 16^x+9^x+1 \ge 3\sqrt [3]{144^x} = 3$ as equality occurs when $16^x=9^x=1$ or $x=0$ .

When $3^x+4^x+12^x - 16^x-9^x = 1$ , $\implies$ LHS $=$ RHS, and LHS equals RHS only when they are minimum at $x=0$ because for $x < 0$ RHS is decreasing faster than LHS and for $x > 0$ RHS is increasing faster then LHS.