JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Consider the function \(\mathrm{f}:(0,2) \rightarrow \mathrm{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function\( g(x)\) defined by \(g(x)=\left\{\begin{array}{cc}\min \{f(t)\}, & 0 < t \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x< 2\end{array}\right.\). Then
- A \(g\) is continuous but not differentiable at \(x=1\)
- B \(\mathrm{g}\) is not continuous for all \(\mathrm{x} \in(0,2)\)
- C \(g\) is neither continuous nor differentiable at \(x=1\)
- D \(g\) is contimuous and differentiable for all \(\mathrm{x} \in(0,2)\)
Answer & Solution
Correct Answer
(A) \(g\) is continuous but not differentiable at \(x=1\)
Step-by-step Solution
Detailed explanation
\( f:(0,2) \rightarrow R ; f(x)=\frac{x}{2}+\frac{2}{x} \) \( f^{\prime}(x)=\frac{1}{2}-\frac{2}{x^2}\) \(\therefore \mathrm{f}(\mathrm{x})\) is decreasing in domain.
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