Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f(x)=\begin{cases}e^{x^2}+x^3+1 & x>0\\ ax^2+bx+2 & x\le 0\end{cases}\)</span> is differentiable at <span class="math-inline">\(x=0\)</span>. Then:</p>
a\in R & b=0
a\in R & b=1
a\in R & b=2
a\in R & b=3
Step-by-Step Solution
Key Concept: General
<div class="solution"><p><strong>Continuity at x=0:</strong> <span class="math-inline">\(f(0^+)=e^0+0+1=2,\ f(0)=2\)</span>. ✓ (Always continuous regardless of a,b.)</p><p><strong>Differentiability:</strong> <span class="math-inline">\(f'(0^+)=2x e^{x^2}+3x^2\big|_0=0\)</span>. <span class="math-inline">\(f'(0^-)=b\)</span>. Need <span class="math-inline">\(b=0\)</span>.</p><p><span class="math-inline">\(a\)</span> can be any real number.</p><p><strong>Answer: (A) a∈R & b=0</strong></p><div class="key-concept"><strong>Key Concept:</strong> Differentiability at a point: match derivatives from both sides</div></div>
Correct Answer: 1