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\( f(x)=\min \phi(t),-3 \leq t \leq x \) where \( \phi(x)=\| x-1|-| x+1|| \) then (1) \( f(x) \) is non-differentiate at \( x=0,-1 \) (2) \( f(x) \) is non-differentiate at \( x=-1,1 \) (3) \( f(100)=0 \) (4) \( \int_{-3}^{10} f(x) d x=5 \) |
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Answer» \(\phi\)(x) = ||x - 1| - |x + 1|| \(=\begin{cases}1-(x-1)-(-(x+1))&;x\leq-1\\1-(x-1)-(x+1)&;-1\leq x\leq1\\|x-1-(x+1)|&;x\geq1\end{cases}\) \(=\begin{cases}2&;x\leq-1\\2|x|&;-1\leq x\leq 1\\2&;x\geq 1\end{cases}\) Now, f(x) = min\(\phi\)(f) ; -3 \(\leq t\leq x\) \(=\begin{cases}2&;-3\leq x\leq-1\,and\,x\geq 1\\0&;-1\leq x\leq1\end{cases}\) \(\left(\because Min |t| = 0\\for\,-1\leq t\leq x \leq 1\right)\) \(\therefore\) f(x) is not differentiable at x = -1 and x = 1 (C) f(100) = 2 (\(\because 100>1\)) (D) \(\int\limits_{-3}^{10}f(x)dx=\int\limits_{-3}^{-1}2dx+\int\limits_{-1}^10dx+\int\limits_1^{10}2dx\) \(=2(x)^{-1}_{-3}+0+2(x)_1^{10}\) = 2(-1-(-3)) + 2(10 - 1) = 2(-1 + 3) + 2 x 9 = 4 + 18 = 22 |
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