A) h is continuous for all x
B) h is differentiable for all x
C) \[h'(x)=1\], for all \[b=1\]
D) h is not differentiable at two values of x
Correct Answer: A
Solution :
\[x\le {{x}^{2}}\,\Rightarrow \,\,x\,(1-x)\le 0\]\[\Rightarrow \,\,x\,(x-1)\ge 0\] \[\Rightarrow \,\,x\le 0\] or \[x\ge 1\]; \[\therefore \,\,\,h(x)=\left[ \begin{array}{*{35}{r}} x\,\,\,\,: & x\le 0 \\ {{x}^{2}}: & 0<x<1 \\ x\,\,\,\,: & x\ge 1 \\ \end{array} \right.\] \[h\,\,(x)\] is continuous for every x but not differentiable at \[x=0\] and 1. Also \[{h}'(x)=\left[ \begin{array}{*{35}{r}} 1 & x<0 \\ \text{not exists} & x=0 \\ 2x & 0<x<1 \\ \text{not exists} & x=1 \\ 1 & x>1 \\ \end{array} \right.\] \[\therefore \,\,\,\,{h}'(x)=1\]for all\[x>1\].
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