Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f(x)=\begin{cases}a+\sin(\sin x) & x\ge 0\\ \ln(\cos x)+bx & x<0\end{cases}\)</span> differentiable at <span class="math-inline">\(x=0\)</span>. Then:</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p><strong>Continuity:</strong> f(0⁺)=a+sin(0)=a; f(0⁻)=ln(1)+0=0. Need a=0. (C) a=0 ✓</p><p><strong>Differentiability:</strong> RHD=cos(sin x)·cos x|₀=1·1=1. LHD=(-sin x/cos x)+b|₀=0+b=b. Need b=1. (D) b=1 ✓</p><p>(A) a=1: FALSE. (B) b=0: FALSE.</p><p><strong>Answer: (C),(D) i.e., a=0 and b=1</strong></p><div class="key-concept"><strong>Key Concept:</strong> Chain rule for derivative of sin(sin x) at x=0</div></div>
Correct Answer: C,D