Applications of Derivatives
Absolute Maxima and Minima
Grade 12
Question:
<p>The function \(x^4 - 4x + 1\) will have</p>
<p>(a) absolute maximum value</p>
<p>(b) absolute minimum value</p>
<p>(c) both absolute maximum and minimum values</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: A fourth-degree polynomial with positive leading coefficient has an absolute minimum but no absolute maximum.
<p>Let f(x) = x⁴ - 4x + 1</p><p>f'(x) = 4x³ - 4 = 4(x³ - 1) = 0 gives x = 1</p><p>f''(x) = 12x² > 0 for all x, so x = 1 is a local minimum</p><p>As x → ±∞, f(x) → +∞ (since x⁴ dominates)</p><p>Therefore, f has an absolute minimum value at x = 1: f(1) = 1 - 4 + 1 = -2</p><p>The function has no absolute maximum</p><p>∴ Answer is (b) absolute minimum value.</p>
Correct Answer: b