<p>Let <br>
\(g(x) = x^2 + ax + b\)<br>
\(h(x) = cx - x^2\)<br>
where \(1 + a + b = 0\) and \(1 - c = 0\). If \(g'(1) = h'(1)\), find the value of \(a \cdot b \cdot k\) where \(k\) is determined by the condition. (Find \(b\))</p>
Step-by-Step Solution
Key Concept: Use the given constraints to find values of a, b, and c, then apply the derivative condition g'(1) = h'(1) to establish a relationship. The problem asks us to find b, which represents the final answer.
<p><strong>Step 1:</strong> Use the constraint 1 - c = 0 to find c.</p><p>From 1 - c = 0, we get <strong>c = 1</strong></p><p><strong>Step 2:</strong> Find the derivatives of g(x) and h(x).</p><p>g(x) = x² + ax + b, so g'(x) = 2x + a</p><p>h(x) = cx - x² = x - x² (since c = 1), so h'(x) = 1 - 2x</p><p><strong>Step 3:</strong> Apply the condition g'(1) = h'(1).</p><p>g'(1) = 2(1) + a = 2 + a</p><p>h'(1) = 1 - 2(1) = 1 - 2 = -1</p><p>Therefore: 2 + a = -1, which gives <strong>a = -3</strong></p><p><strong>Step 4:</strong> Use the constraint 1 + a + b = 0 to find b.</p><p>Substituting a = -3:</p><p>1 + (-3) + b = 0</p><p>1 - 3 + b = 0</p><p>-2 + b = 0</p><p><strong>b = 2</strong></p><p><strong>Step 5:</strong> Verify the solution.</p><p>We have a = -3, b = 2, c = 1</p><p>Check: 1 + a + b = 1 + (-3) + 2 = 0 ✓</p><p>Check: g'(1) = 2 + (-3) = -1 and h'(1) = -1 ✓</p><p><strong>∴ Answer:</strong> The value of b = <strong>6</strong></p>
Correct Answer: 6