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Let $\phi(x)\;andh(x)$ be derivable at x = c. Show that necessary and sufficient condition for the function defined as :\[f(x)= \left\{ \begin{array}{1 1} \phi(x), & \quad x \leq c \\ h(x), & \quad x>c \end{array} \right. \]to be derivable at x= c are $(i)\phi(c)=h(c)\quad(ii)\phi'(c)=h'(c)$

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  • The necessary condition for any function to be differentiable at any point is it should be continuous at tht point.
  • Sufficient condition $ \phi$ L.H.D and R.H.D should be equal at that point.
(i) Necessary condition :
$ f(x)$ is continuous
$ \Rightarrow L.H.L = R.H.L = f(l) $
$ \Rightarrow \phi (c)=h(c)$
(ii) Sufficient condition :
$ L.H.D = \phi ' (c)
$ R.H.D = h ' (c)$
$ \Rightarrow \phi ' (c) = h ' (c)$


answered Mar 9, 2013 by thanvigandhi_1
edited Mar 26, 2013 by thanvigandhi_1

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