Basic Mathematics & Logarithm
Inequalities and Optimization
Grade 11
Question:
<p>If <i>x</i>, <i>y</i>, <i>z</i> are positive and <i>x</i> + <i>y</i> + <i>z</i> = 1, then the minimum value of <i>x</i><sup>-1</sup> + <i>y</i><sup>-1</sup> + <i>z</i><sup>-1</sup> is</p>
<p>(A) 12</p>
<p>(B) 4</p>
<p>(C) 8</p>
<p>(D) 9</p>
Step-by-Step Solution
Key Concept: Use Cauchy-Schwarz inequality to relate the sum of reciprocals with the original constraint.
<p><strong>Solution:</strong> By Cauchy-Schwarz inequality, <i>x</i> + <i>y</i> + <i>z</i> multiplied by <i>x</i><sup>−1</sup> + <i>y</i><sup>−1</sup> + <i>z</i><sup>−1</sup> ≥ (1 + 1 + 1)<sup>2</sup> = 9.</p><p>Since <i>x</i> + <i>y</i> + <i>z</i> = 1, we have <i>x</i><sup>−1</sup> + <i>y</i><sup>−1</sup> + <i>z</i><sup>−1</sup> ≥ 9.</p><p>Equality holds when <i>x</i> = <i>y</i> = <i>z</i> = 1/3.</p><p>∴ Answer is D.</p>
Correct Answer: D