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if f(x) = { 3-x2 , -1[_x[_2

{ 2x-4 ,  2[_x[_4 then find fof(x)and discuss the continuty and differentiability

4 years ago

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Ans:
$f(x) = \left \{ 3-x^{2}, -1\leq x< 2 \right \}$
$f(x) = \left \{ 2x-4, 2\leq x\leq 4 \right \}$
Continuity,
Left Hand Limit(LHL) = Right Hand Limit(RHL) = f(2)
$LHL = \lim_{x\rightarrow 2^{-}}3-x^{2} = -1$
$RHL = \lim_{x\rightarrow 2^{+}}2x-4 = 1$
$LHL\neq RHL$
f(x) is not continuous at x = 2
Similarly you can find Left Hand Derivetive & Right Hand Derivative
$LHD=\lim_{x\rightarrow 2^{-}}\frac{f(x+2)-f(2)}{x}$
$RHD=\lim_{x\rightarrow 2^{+}}\frac{f(x+2)-f(2)}{x}$
f(x) is not differentiable at x = 2
Thanks & Regards
Jitender Singh
IIT Delhi
2 years ago

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