1.

Show that the function `f(x)=x^3/2`is not differentiable at x=0.

Answer» `f(x)=x^(3//2)`
`R,f (0)=underset(hrarr0)lim f(0+h)-f(0))/(h)`
`=underset(hrarr0)lim(h^(3//2)-0)/(h)=underset(hrarr0) limh^(1//2)=0`
and L.f(0) L.f(0)`=underset(hrarr0)lim (f(0-h)-f(0))/((-h))`
`=underset(hrarr0)lim(h^(3//2)-0)/(-h)`
`=underset(hrarr0)lim((-h)^(3//2)-0)/(-h)`
`=underset(hrarr)lim^(1//2)` which is imaginary .
`therefore` L.f (0) does not exist
Therefore f(x) is not differentiable at x=0.


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