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Question

Let f(x)=sin|x2-5x+6|x2-5x+6x2,31x=2or3 , the set of all points where f is differentiable is


A

-,

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B

-,-2

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C

-,-3

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D

-,-2,3

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Solution

The correct option is D

-,-2,3


Explanation for Correct Answer:

Use limits to find the differentiable points:

Given that:

f(x)=sin|x2-5x+6|x2-5x+6x2,31x=2or3

The sine function is always differentiable. Also, 1 is constant and hence always differentiable.

Hence the function may or may not be differentiable at only two points that are 2,3.

Let's check at x=2.

f'2+=limx0f(2+h)+f(2)h=limh0h(h-1)h=limh0sin(h-1)h2(h-1)=-

Let's check at x=3.

f'3+=limx0f(3+h)+f(3)h=limh0h(h-1)h=limh0sin(h-1)h2(h-1)=-

We can say that x is not differentiable at x=2and3.

Therefore, The required set is -,-2,3

Therefore, the correct answer is option (D).


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