Applications of Derivatives
Maxima and Minima with Constraints
Grade 12
Question:
<p>The possible value of <i>|a - c|</i>, if <i>a</i><sup>2</sup> + 2<i>b</i><sup>2</sup> is maximum, is given by</p>
<p>(a) 1</p>
<p>(b) 0</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Use the constraint |f(x)| ≤ 1 for |x| ≤ 1 to bound the coefficients and then optimize a² + 2b².
<p>Using the constraint condition from the passage that |f(x)| ≤ 1 for |x| ≤ 1, we apply:</p><p>|f(1) - f(-1)| ≤ 2 and |f(1) - f(0)| ≤ 2</p><p>This gives us |(a + b + c) - (a - b + c)| ≤ 2, so |2b| ≤ 2</p><p>And |f(-1) - f(0)| ≤ 2 gives |a - b + c - c| ≤ 2, so |a - b| ≤ 2</p><p>From the optimization with maximum a² + 2b², we find |a - c| = 2</p><p>∴ Answer is (c) 2.</p>
Correct Answer: c