Limits, Continuity & Differentiability
Higher Order Derivatives
Grade 12
Question:
<p>If <span>\(f(x) = (x-1)^4(x-2)^3(x-3)^2\)</span>, then the value of <span>\(f'''(1) + f''(2) + f'(3)\)</span> is:</p>
<p>(a) <span>\(0\)</span></p>
<p>(b) <span>\(1\)</span></p>
<p>(c) <span>\(2\)</span></p>
<p>(d) <span>\(6\)</span></p>
Step-by-Step Solution
Key Concept: When a function has a factor (x-a)ⁿ, then the first n-1 derivatives at x=a are zero.
<p><strong>Step 1:</strong> At <span>$x=1$</span>: <span>$f(x) = (x-1)^4(x-2)^3(x-3)^2$</span> has a factor <span>$(x-1)^4$</span>, so <span>$f(1) = f'(1) = f''(1) = f'''(1) = 0$</span>.</p><p><strong>Step 2:</strong> At <span>$x=2$</span>: <span>$f(x)$</span> has a factor <span>$(x-2)^3$</span>, so <span>$f(2) = f'(2) = f''(2) = 0$</span>.</p><p><strong>Step 3:</strong> At <span>$x=3$</span>: <span>$f(x)$</span> has a factor <span>$(x-3)^2$</span>, so <span>$f(3) = f'(3) = 0$</span>.</p><p>Therefore, <span>$f'''(1) + f''(2) + f'(3) = 0 + 0 + 0 = 0$</span>.</p>
Correct Answer: A