a.b=?

$Arithmetic\quad Mean\quad \ge \quad Geometric\quad Mean\\ \frac { a\quad +\quad b }{ 2 } \quad \ge \quad \sqrt { ab } \\ Square\quad Both\quad Sides\\ \Longrightarrow \quad (a\quad +\quad b\quad )^{ 2 }\quad \ge \quad ab\\ \Longrightarrow \quad \frac { (20)^{ 2 } }{ 4 } \quad \ge \quad ab\\ \twoheadrightarrow \quad \frac { 400 }{ 4 } \ge \quad ab\quad \\ Hence\quad 100\quad \ge \quad ab$

$a + b = 100$ . Let $a = b = 10$ . So, maximum value of $a × b = 10 × 10 = \color{#69047E}{\boxed{100}}$ .