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Question

A function f is defined by f(x)=2+(x-1)23 in [0,2]. Which of the following is not correct?


A

f is not derivable (0,2)

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B

f is not continuous in [0,2]

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C

f(0)=f(2)

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D

Rolle's theorem is correct in[0,2]

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Solution

The correct option is D

Rolle's theorem is correct in[0,2]


The explanation for the correct option:

Step 1: Finding the differentiability:

Given, the function, f(x)=2+(x-1)23 in 0,2

f'x=0+23×1x-113=23×1x-113

At x=1,

f'1=23×11-113=

Thus, f(x) is not differentiable.

Step 2: Finding the continuity of the function:

At x=1,f(1)=2+(1-1)23

=2

Thus, f is continuous in [0,2].

Step 3: Finding f(0)and f(2):

f(0)=2+(0-1)23=2+1=3

f(2)=2+(2-1)23=2+1=3

Thus f0=f2

Step 4: Check the function to satisfy Rolle's theorem:

For the function to satisfy Rolle's theorem

It should satisfy the three conditions that is differentiability, continuity and f(0)=f(2).

But it does not satisfies all the three condition.

Thus, the function does not satisfy Rolle's theorem at [0,2].

Hence option (C),(D) is the correct answer.


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