Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(g(x)=\begin{cases}x+b & x<0\\ \cos x & x\ge 0\end{cases}\)</span> can be made differentiable at <span class="math-inline">\(x=0\)</span> if:</p>
if b is zero
if b is not zero
if b takes any real value
<strong>for no value of b</strong>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p><strong>Continuity:</strong> <span class="math-inline">$g(0^-)=b,\ g(0)=1$</span>. Need <span class="math-inline">$b=1$</span>.</p><p><strong>Differentiability with b=1:</strong> <span class="math-inline">$\text{LHD}=1,\ \text{RHD}=-\sin(0)=0$</span>. LHD≠RHD, so <strong>not differentiable for any b</strong>.</p><p><strong>Answer: (D) for no value of b</strong></p><div class="trap-box"><strong>Trap:</strong> Students set b=1 for continuity and assume differentiability follows automatically.</div><div class="key-concept"><strong>Key Concept:</strong> Continuity is necessary but not sufficient for differentiability</div></div>
Correct Answer: 4