Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p>Let <em>f</em>(<em>x</em>) = (<em>x</em> − 5)<sup>55</sup> (<em>x</em> − 6)<sup>66</sup>. Find the number of points of local maxima, local minima, and points of inflexion, and identify the nature of critical points.</p><p>Specifically, at <em>x</em> = 5, 6, and <em>x</em> = 660/121, what is the nature of each critical point?</p>
<p>x = 5 is point of inflexion, x = 660/121 is local maximum, x = 6 is local minimum</p>
<p>x = 5 is point of inflexion, x = 660/121 is local minimum, x = 6 is local maximum</p>
<p>x = 5 is local minimum, x = 660/121 is local maximum, x = 6 is local minimum</p>
<p>x = 5 is local maximum, x = 660/121 is local minimum, x = 6 is local maximum</p>
Step-by-Step Solution
Key Concept: At a critical point where f'(x) = 0, determine the nature by analyzing the sign change of f'(x) across that point and the multiplicity of roots in f'(x). For points where a factor has odd power in f(x), f'(x) changes sign (local extremum); for even powers, f'(x) doesn't change sign (inflection point).
<p><strong>Step 1: Find f'(x)</strong></p><p>f(x) = (x−5)^55(x−6)^66</p><p>f'(x) = 55(x−5)^54(x−6)^66 + 66(x−5)^55(x−6)^65</p><p>f'(x) = (x−5)^54(x−6)^65[55(x−6) + 66(x−5)]</p><p>f'(x) = (x−5)^54(x−6)^65[55x − 330 + 66x − 330]</p><p>f'(x) = (x−5)^54(x−6)^65[121x − 660]</p><p><strong>Step 2: Find critical points</strong></p><p>Critical points: x = 5, x = 6, x = 660/121</p><p><strong>Step 3: Analyze sign change of f'(x)</strong></p><p><strong>At x = 5:</strong> Factor (x−5)^54 has even power (54). The sign of f'(x) does NOT change as x passes through 5. Since f(5) = 0 and f(x) ≥ 0 nearby (product of even power), <strong>x = 5 is a point of inflection</strong> (horizontal inflection).</p><p><strong>At x = 6:</strong> Factor (x−6)^65 has odd power (65). The sign of f'(x) DOES change at x = 6. Testing intervals: f'(x) > 0 for x < 6 and f'(x) < 0 for x > 6 (after accounting for all factors). Thus <strong>x = 6 is a local maximum</strong>.</p><p><strong>At x = 660/121 ≈ 5.455:</strong> Factor (121x−660) has power 1 (odd). The sign of f'(x) changes from negative to positive. Thus <strong>x = 660/121 is a local minimum</strong>.</p><p><strong>Summary:</strong> 1 local minimum, 1 local maximum, 1 inflection point.</p><p>∴ Answer: A</p>
Correct Answer: A