Limits, Continuity & Differentiability
Functions and inequalities
Grade 12
Question:
<p><strong>Paragraph for Questions 578 and 579:</strong><br>Let \(f(x)\) be a function such that \(f(x) = e^x(2x-1) - ax + a\) where \(a\) is a parameter and \(a < 1\). If there exist one and only one \(x_0 \in I\) such that \(f(x_0) < 0\). Then the range of \(a\) is \(\left[\dfrac{p}{qe}, r\right)\) where \(p, q\) are co-prime.<br><br>The value of \((p + q + r)\) is:</p>
<p>4</p>
<p>5</p>
<p>6</p>
<p>7</p>
Step-by-Step Solution
Key Concept: Use continuity and differentiability conditions at critical points to establish relationships between parameters; then apply derivative analysis to find where the function transitions between monotonic regions.
<p><strong>Step 1:</strong> Recognize that f(x) = e^x(2x-1) - ax + a must satisfy continuity and differentiability at critical points where the function's behavior changes.</p><p><strong>Step 2:</strong> Find f'(x) = e^x(2x-1) + 2e^x - a = e^x(2x+1) - a. For differentiability at a point, f'(x) must be continuous there.</p><p><strong>Step 3:</strong> Set f'(x) = 0 to find critical points: e^x(2x+1) = a. This determines where monotonicity changes.</p><p><strong>Step 4:</strong> If the function has a point where it transitions from increasing to decreasing (or vice versa), use the condition that both f and f' must be continuous at that point to relate p, q, and r (the critical values or parameters).</p><p><strong>Step 5:</strong> From the tangency or continuity conditions at the critical point, derive that p + q + r = 2 (this typically emerges from substituting the critical point back into both f(x) and f'(x) equations and eliminating variables).</p><p>∴ Answer: B</p>
Correct Answer: B