Show that f(x)= x² is
differentiable in 0≤ x ≤ 2.
Please solve this step by step
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Answer:
Left hand limit of f(x) at x=1 →lim
h→0
(1−h)=1,
Right hand limit of f(x) at x=1 →lim
h→0
(2−(1+h))=lim
x→0
(1−h)=1
Since LHL=RHL
So, f(x) is continuous at x=1.
Left hand derivative of f(x) at x=1 →lim
h→0
(1−h)−1
(1−h)−1
=1,
Right hand derivative of f(x) at x=1 →lim
x→0
(1+h)−1
{2−(1+h)}−(2−1)
=−1,
Since, LHD
=RHD
So, f(x) is not differentiable at x=1.
Left hand limit of f(x) at x=2 →lim
h→0
(2−(2−h))=0,
Right hand limit of f(x) at x=2 →lim
h→0
(−2+3(2+h)−(2+h)
2
)=0
Since LHL=RHL
So, f(x) is continuous at x=2.
Left hand derivative of f(x) at x=2 →lim
h→0
(2−h)−2
{2−(2−h)}−(2−2)
=−1,
Right hand derivative of f(x) at x=2 →lim
x→0
(2+h)−2
{−2+3(2+h)−(2+h
2
)}−(−2+3(2)−2
2
)
=−1,
Since, LHD=RHD
So, f(x) is differentiable at x=2.
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