Question

Show that f(x)= x² is

differentiable in 0≤ x ≤ 2.


Please solve this step by step​

Answer 1

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.