Maths · Limits, continuity and differentiability
Let \) be defined in the interval [-2,2] such that = \) =\right. \) +|\boldsymbo
Let \( \boldsymbol{f}(\boldsymbol{x}) \) be defined in the interval [-2,2] such that \( f(x)= \) \( \left\{\begin{array}{ll}-1, & -2 \leq x \leq 0 \\ x-1, & 0<x \leq 2\end{array} \text { and } g(x)=\right. \) \( \boldsymbol{f}(|\boldsymbol{x}|)+|\boldsymbol{f}(\boldsymbol{x})| \) Test the differentiablity of \( g(x) \) in (-2,2)
- A. not derivable at \( x=0 \) and \( x=1 \)
- B. derivable at all points
- C. not derivable at \( x=0 \)
- D. not derivable at \( x=1 \)
Step-by-step solution
We define g(x) piecewise. For x<0, g(x) = -x. For 0<x<1, g(x) = 0. For 1<x<2, g(x) = 2x-2. At x=0, left derivative = -1, right derivative = 0, so not differentiable. At x=1, left derivative = 0, right derivative = 2, so not differentiable. Hence g is not derivable at x=0 and x=1.
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