COMEDK · Maths · 25. Continuity and Differentiability
If \(f(x)=\left\{\begin{array}{ll}a x+3, & x \leq 2 \\ a^{2} x-1 & x>2\end{array}\right.\), then the values of \(a\) for which \(f\) is continuous for all \(x\) are
- A 1 and \(-2\)
- B 1 and 2
- C \(-1\) and 2
- D \(-1\) and \(-2\)
Answer & Solution
Correct Answer
(C) \(-1\) and 2
Step-by-step Solution
Detailed explanation
For the function \(f(x)\) to be continuous for all \(x\), it must be continuous at the point \(x = 2\). The condition for continuity at \(x = 2\) is \(\lim_{x \to 2^{-}} f(x) = \lim_{x \to 2^{+}} f(x) = f(2)\). Calculating the left-hand limit at \(x = 2\):…
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