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Examine the differentiability of f, ...

Examine the differentiability of f, where f is defined by `f(x)={{:(1+x, if x le 2),(5-x,ifx gt 2):}` at `x = 2`.

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We have, `f(x)={{:(1+x, if x le 2),(5-x,ifx gt 2):}` at `x = 2`.
For differentiability at `x = 2`.
`Lf'(2) = underset(xrarr2^(-))(lim)(f(x)-f(2))/(x-2)=underset(xrarr2^(-))(lim)((1+x)-(1+2))/(x-2)`
`= underset(hrarr0)(lim)((1+2-h)-3)/(2-h-2) = underset(hrarr0)(lim)(-h)/(h) = 1`
`Rf'(2) = underset(xrarr2^(+))lim(f(x)-f(2))/(x-2)=underset(xrarr2^(+))lim((5-x)-3)/(x-2)`
`=underset(hrarr0)(lim)(5-(2+h)-3)/(2+h-2)`
`= underset(hrarr0)(lim) (5-2-h-3)/(h) = underset(hrarr0)(lim)(-h)/(+h)`
`= - 1`
`:' Lf'(2) = ne Rf'(2)`
So, `f(x) ` is not differentiable at `x = 2`.
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