Applications of Derivatives
Extrema conditions
Grade 12
Question:
<p>From the function \( f \), we have \( f'(x) = \dfrac{1}{x} + 2bx + a \). It is given that \( f \) has extreme values and hence differentiable. Which of the following is a condition that must hold?</p>
<p>\( a^2 - 8b > 0 \)</p>
<p>\( a^2 - 8b = 0 \)</p>
<p>\( a^2 - 8b < 0 \)</p>
<p>\( a^2 + 8b > 0 \)</p>
Step-by-Step Solution
Key Concept: For f to have extreme values, f'(x) = 0 must have at least two distinct real roots. This requires the discriminant of the quadratic equation (in terms of x) formed by f'(x) = 0 to be positive, which imposes a constraint on parameters a and b.
<p><strong>Step 1:</strong> For f to have extreme values, f'(x) must have at least two distinct zeros where it changes sign (local maximum and minimum).</p><p><strong>Step 2:</strong> Set f'(x) = 0: $\frac{1}{x} + 2bx + a = 0$</p><p>Multiply by x (assuming x ≠ 0): $1 + 2bx^2 + ax = 0$, or $2bx^2 + ax + 1 = 0$</p><p><strong>Step 3:</strong> For two distinct real roots, the discriminant must be positive:</p><p>$\Delta = a^2 - 4(2b)(1) > 0$</p><p>$a^2 - 8b > 0$</p><p>∴ The condition that must hold is: <strong>$a^2 > 8b$</strong> (or equivalently $b < \frac{a^2}{8}$)</p>
Correct Answer: A