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This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 2 based on use of Sandwich Theorem . Let's give it a try !!

Let f be a real valued function satisfying for some and

(a) If show that f is continuous at a

(b) If show that f is differentiable at a

Differentiability

Continuity

Limit

Sandwich Theorem

(a) We are given that for some .

We have to show that f is continuous at x=a . For this it's enough to show that .

Now taking limit we have ,

Using Sandwich theorem we can say that . Since

Hence f is continuous at x=a proved .

(b) Here we have to show that f is differentiable at x=a for this it's enough to show that the exists .

We are given that , for some and ,

which implies

Now taking we get by Sandwich theorem i.e f'(a)=0 .

Since , , for .

Hence f is differentiable at x=a proved .

be such that for some and all . Show that f must have a unique fixed point .

This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 2 based on use of Sandwich Theorem . Let's give it a try !!

Let f be a real valued function satisfying for some and

(a) If show that f is continuous at a

(b) If show that f is differentiable at a

Differentiability

Continuity

Limit

Sandwich Theorem

(a) We are given that for some .

We have to show that f is continuous at x=a . For this it's enough to show that .

Now taking limit we have ,

Using Sandwich theorem we can say that . Since

Hence f is continuous at x=a proved .

(b) Here we have to show that f is differentiable at x=a for this it's enough to show that the exists .

We are given that , for some and ,

which implies

Now taking we get by Sandwich theorem i.e f'(a)=0 .

Since , , for .

Hence f is differentiable at x=a proved .

be such that for some and all . Show that f must have a unique fixed point .

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