x, y, z > 0 if xy z = 8, then least value of x + y + z is?
Understand the Problem
The question is asking for the least value of the expression x + y + z, given that the variables x, y, and z are all greater than zero and the product xy z equals 8. This involves finding the minimum of the sum under the given constraint, typically solved using techniques like the AM-GM inequality or Lagrange multipliers.
Answer
The least value of $x + y + z$ is $6$.
Answer for screen readers
The least value of $x + y + z$ is $6$.
Steps to Solve
- Applying the AM-GM Inequality
To minimize the expression $x + y + z$ given the constraint $xyz = 8$, we utilize the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for any non-negative numbers $a_1, a_2, \ldots, a_n$, the following holds: $$ \frac{a_1 + a_2 + \ldots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n} $$ Here, we use $a_1 = x$, $a_2 = y$, and $a_3 = z$.
- Setting Up AM-GM
For our numbers, we have: $$ \frac{x + y + z}{3} \geq \sqrt[3]{xyz} $$ Given that $xyz = 8$, we find: $$ \sqrt[3]{8} = 2 $$
- Calculating the Minimum Sum
Substituting back, we find: $$ \frac{x + y + z}{3} \geq 2 $$ Multiplying both sides by 3 gives: $$ x + y + z \geq 6 $$
- Finding When Equality Holds
The equality in the AM-GM inequality holds when $x = y = z$. Hence, we set $x = y = z = 2$, which satisfies the constraint: $$ xyz = 2 \cdot 2 \cdot 2 = 8 $$
The least value of $x + y + z$ is $6$.
More Information
The result utilizes the AM-GM inequality, a powerful mathematical principle that states the arithmetic mean is always greater than or equal to the geometric mean. The minimum sum occurs when $x$, $y$, and $z$ are equal.
Tips
- Not applying the AM-GM inequality correctly, or miscalculating the geometric mean can lead to incorrect conclusions.
- Failing to verify that the values satisfy the original constraint can result in invalid answers.
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