check the fn=|x| is continuous and diferrentiable at x=2
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Answer:
Given f(x)=2x∣x∣
we know that ∣x∣=x∀ x>0,∣x∣=−x ∀ x<0
then
f(x)=
−2x
2
2x
2
x<0
x>0
f
1
(x)=
−4x
4x
x<0
x>0
f(x) is differentiable everywhere except x=0
So,
f
1
(0
−
)=0
f
1
(0
+
)=0
∵f
1
(0
−
)=f
1
(0
+
) so f(x) is differentiable at x=0
Whice mean f(x) is differentiable in (−∞,∞)
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