Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f(x)=x^3-x^2+(x-1)\sin x\)</span>. <span class="math-inline">\((fg)(x)=f(x)g(x)\)</span>. Which TRUE?</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f(1)=0. (A) g cont at 1 ⟹ fg diff at 1 ✓. (C) g diff at 1 ⟹ fg diff at 1 ✓. (B),(D) FALSE: fg diff does not force g to be cont/diff. <strong>Answer: (A),(C)</strong></p><div class="key-concept"><strong>Key Concept:</strong> When f(a)=0: (fg)'(a)=f'(a)g(a) requires only g(a) exists, i.e., g continuous</div></div>
Correct Answer: A,C