Continuity And Differentiability

Step 1: Concept
.
Step 2: Analysis
Let and .
and .
Step 3: Conclusion
.
Final Answer: (B)

Step 1: Concept
Write the function as and square both sides.
Step 2: Analysis
. Differentiating both sides with respect to : .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
Examine the limit and the derivative of the absolute value function at the origin.
Step 2: Analysis
, so it is continuous. Left-hand derivative is and right-hand derivative is .
Step 3: Conclusion
Since the left and right derivatives are not equal, the function is not differentiable at .
Final Answer: (B)

Step 1: Concept
Apply the Chain Rule: .
Step 2: Analysis
Let . Then . .
Step 3: Conclusion
Substituting back: .
Final Answer: (A)

Step 1: Concept
Apply the Chain Rule: .
Step 2: Analysis
Let . Then . .
Step 3: Conclusion
Substituting back: .
Final Answer: (A)

Step 1: First and Second Derivatives
. Differentiating again: . Dividing by : .
Step 2: Successive Differentiation (Leibnitz Rule)
Differentiating times: . .
Step 3: Conclusion
.
Final Answer: (c)

Step 1: Express the function .
We are given the function: To find the second partial derivatives of , we will differentiate with respect to and .

Step 2: Differentiate with respect to .
First, compute the partial derivative of with respect to : Now, take the second derivative with respect to :

Step 3: Differentiate with respect to .
Next, compute the partial derivative of with respect to : Now, take the second derivative with respect to :

Step 4: Compare and .
From the expressions for and , we observe: Thus, the correct answer is , which corresponds to option (C).

Concept: Non-differentiability occurs where modulus argument is zero or function is non-smooth.

Step 1: Solve inside modulus.

Step 2: Check smoothness: At : At :

Step 3: is differentiable everywhere except possibly at 0, but it is smooth at 0. Final Answer:

Concept: Functional equation of exponential type:

Step 1: Use derivative at 0.


Step 2: Find .


Step 3: Compute derivative.


Step 4: Evaluate using given relation.
Conclusion:

Concept: Infinite nested expression set equal to whole.

Step 1: Form equation.


Step 2: Square.


Step 3: Differentiate.


Step 4: Solve.

Concept: A function equals its derivative if .

Step 1: Check each option.

Step 2: Conclusion. Only remains unchanged after differentiation.