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Question
Given f (x) = 2x, x < 0
= 0, x ≥ 0
then f (x) is _______ .
Options
discontinuous and not differentiable at x = 0
continuous and differentiable at x = 0
discontinuous and differentiable at x = 0
continuous and not differentiable at x = 0
Solution
continuous and not differentiable at x = 0
solution:
f (x) = 2x, x < 0
= 0, x ≥ 0
`lim_(x->0^-)f(x)=lim_(x->0^-)2x=0`
`lim_(x->0^+)f(x)=lim_(x->0^+)0=0`
and f(0) = 0
`lim_(x->0^-)f(x)=lim_(x->0^+)f(x)=f(0)`
Hence, f(x) is continuous at x = 0.
Now we find left hand derivative and right hand derivative of f(0) at x = 0
Right hand derivative at x = 0
i.e `f'(0^+)=lim_(h->0^+)(f(0+h)-f(0))/h=lim_(h->0^+)(0-0)/h=0`
Left hand derivative at x = 0
i.e `f'(0^-)=lim_(h->0^-)(f(0+h)-f(0))/h=lim_(h->0^+)(h-0)/h=1`
`f'(0^+)ne f'(0^-)` Hence, f(x) is not differentiable at x = 0.
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