Maths · Limits, continuity and differentiability
Let =\boldsymbol{x}^{3}-\boldsymbol{x}^{2}+\boldsymbol{x}+\mathbf{1} \) and = \)
Let \( \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}^{3}-\boldsymbol{x}^{2}+\boldsymbol{x}+\mathbf{1} \) and \( \boldsymbol{g}(\boldsymbol{x})= \) \( \left\{\begin{array}{l}\max \{f(t)\}, \quad 0 \leq t \leq x \quad 0 \leq x \leq 1 \\ 3-x, \quad 1<x \leq 2\end{array}\right. \) Then in the interval \( [0,2], g(x) \) is This question has multiple correct options
- A. Continuous for all \( x \)
- B. Differentiable for all \( x \)
- C. Discontinuous at \( x=1 \)
- D. Not differentiable at \( x=1 \)
Step-by-step solution
f(x) is strictly increasing on [0,1], so g(x)=f(x) on [0,1] and g(x)=3-x on (1,2]. At x=1, g(1)=f(1)=2, and limit from right is 2, so continuous. Left derivative f'(1)=2, right derivative -1, so not differentiable at x=1. Option A is also correct, but the single best correct option is D.
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