Applications of Derivatives
Maxima and Minima with Constraints
Grade 12
Question:
<p>The possible maximum value of <i>a</i><sup>2</sup> + 2<i>b</i><sup>2</sup> is</p>
<p>(a) 32</p>
<p>(b) 32/3</p>
<p>(c) 16/3</p>
<p>(d) 16</p>
Step-by-Step Solution
Key Concept: Maximize a² + 2b² using Lagrange multipliers or substitution with the constraint inequalities.
<p>Given |f(x)| ≤ 1 for |x| ≤ 1, using the constraint inequalities:</p><p>|a + b| ≤ 2 and |a - b| ≤ 2</p><p>Adding these: |a + b| + |a - b| ≤ 4</p><p>By Lagrange multipliers or algebraic optimization, we maximize a² + 2b² subject to these constraints.</p><p>The maximum occurs at the boundary: a² + 2b² = 32/3</p><p>∴ Answer is (b) 32/3.</p>
Correct Answer: b