Kerala KEAM 5 June 2024 Solved Paper

We are given that $abc = 1$ and we need to find the minimum value of $a + b + c$, where $a, b, c$ are positive numbers.Step 1: Use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. The AM-GM inequality states that for positive numbers $x_1, x_2, \dots, x_n$, $$\frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \dots x_n},$$with equality holding if and only if $x_1 = x_2 = \dots = x_n$.Step 2: Apply the AM-GM inequality to the numbers $a, b, c$: $$\frac{a + b + c}{3} \geq \sqrt[3]{abc}.$$Since $abc = 1$, we have: $$\frac{a + b + c}{3} \geq \sqrt[3]{1} = 1.$$Multiplying both sides by 3: $$a + b + c \geq 3.$$Step 3: The equality holds when $a = b = c$. Since $abc = 1$, if $a = b = c$, then $a^3 = 1$, so $a = 1$. Thus, $a = b = c = 1$.Therefore, the minimum value of $a + b + c$ is $1 + 1 + 1 = 3$.Thus, the correct answer is option (E).Quick Tip: The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a powerful tool for finding the minimum or maximum value of a sum under certain conditions. In this case, it helps us determine the minimum value of $a + b + c$ when $abc = 1$.