Continuity And Differentiability

Step 1: For continuity at , Step 2: Left-hand limit (LHL): Right-hand limit (RHL): Step 3: So, Therefore, . %Quciktip

We are given the function . Let's compute the first and second derivatives of . First derivative: Second derivative: After simplification, we get:

We are asked to differentiate with respect to . Let: This simplifies to: Now, differentiate with respect to . Since we are differentiating with respect to , we use the chain rule: This is the required differentiation.

We are given that . Differentiating both sides with respect to , we get: Since is a constant, its derivative is zero. Now, using the chain rule for differentiation: The derivative of with respect to is: Thus, the equation becomes: For this equation to hold, we must have: Solving for :

For to be continuous at , the limit of as approaches 0 must equal the value of at , which is . We first simplify the expression for : Using the logarithmic property, , we get: Now, expand the product inside the logarithm: For small values of , we can ignore the term. Thus, we have: Using the approximation for small , we get: Therefore, for continuity, .

The function is continuous at because the limit of as approaches 0 from both sides equals the function value at . However, is not differentiable at because the left-hand and right-hand derivatives do not match at that point. The derivative is undefined at . Thus, the incorrect statement is that is differentiable at .

We are given: Let . Using the chain rule, the derivative of is: Now, compute : and apply the quotient rule to find: Simplifying the numerator: Now, substitute this into the derivative formula for :

We are asked to differentiate the function: Let . Using the chain rule, the derivative of is: First, compute : Now, apply the chain rule: This is the required derivative.

We are given the equation: Taking the natural logarithm of both sides: Using the logarithmic identity , we get: Now, differentiate both sides with respect to . On the left-hand side, apply the product rule to , and on the right-hand side, apply the product rule to : Differentiating: Now, collect terms involving on one side: Factor out : Finally, solve for :

To determine the correctness of the Assertion (A) and Reason (R), we analyze the greatest integer function , where denotes the greatest integer less than or equal to . - Assertion (A): , , the greatest integer function is not differentiable at .
The greatest integer function has a step-like graph, with jumps at every integer. Differentiability requires the function to have a well-defined derivative, which involves the existence of left-hand and right-hand derivatives being equal. At :
- Left-hand derivative at : As approaches 2 from the left (e.g., ), , so the difference quotient approaches 0.
- Right-hand derivative at : As approaches 2 from the right (e.g., ), , so the difference quotient approaches infinity.
Since the left-hand and right-hand derivatives are not equal, is not differentiable at . Assertion (A) is true.
- Reason (R): The greatest integer function is not continuous at any integral value.
The greatest integer function has discontinuities at integer values because it jumps from one integer to the next. For example, at :
- As approaches 2 from the left, .
- As approaches 2 from the right, .
- The value at is , so the limit does not exist, and the function is not continuous at . This holds for all integers. Thus, Reason (R) is true.
- Explanation Link: Differentiability requires continuity. Since the greatest integer function is not continuous at (and at any integer), it cannot be differentiable at . Reason (R) explains why Assertion (A) is true, as the discontinuity at integral values prevents differentiability. - Conclusion: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). The correct option is (A).

Step 1: Differentiate for .
For the interval : Differentiate using the power rule: Thus Step 2: Find .
For : Differentiate: Thus Step 3: Test continuity at .
A function is continuous at if Left hand limit: Substitute : Right hand limit: Function value: Since the function is continuous at .
Step 4: Test differentiability at .
Compute left derivative. For : Thus Now compute right derivative. For : Thus Since the function is not differentiable at .
Final Answer:
The function is continuous at but not differentiable at .