Differentiability

To determine the number of points where the function is not differentiable, we need to analyze the points of non-differentiability in each term of the function.

The full function can be split into two components:

We shall determine points of non-differentiability in each part and then combine them to find the overall points of non-differentiability for .

Step 1: Analyzing

• The expression involves absolute values of and . Points and are potential points of non-differentiability (because the absolute value function has corners at arguments where the argument becomes zero).

Step 2: Analyzing

• The expression , and it involves the absolute value, impacting points where argument is zero, i.e., and .
• Thus, points of interest for non-differentiability due to absolute values are and , because itself does not make this term non-differentiable when .

Step 3: Conclusion

Combining results from both components, we look at the union of all potential non-differentiable points:

• The points are where absolute values cause non-differentiability.
• Point in does not actually introduce non-differentiability due to the linear factor cancelling out the in the expression .

Thus, the number of points where is not differentiable is 3.

Since , we observe that is symmetric about , suggesting that the behavior around is crucial.

Calculate :

Given , we find:

and similarly,

Calculate :

Since is non-constant and twice differentiable, by the Intermediate Value Theorem, must occur at least twice in .

Given:

The equation is:

Step 1: Rewriting the equation:

First, simplify the terms:

Step 2: Differentiating the equation with respect to :

Now differentiate the equation with respect to : Simplifying:

Step 3: Evaluating at :

Now substitute into : Simplifying:

Step 4: Final Calculation:

Now multiply by 96 to find:

Conclusion: The final result is 105.

To solve the problem, we need to determine the value of given the function and the conditions for continuity and differentiability at specific points.

Step 1: Understand the function and conditions.

• Given:
• is continuous at .
• .

Step 2: Find conditions for continuity at .

• For continuity at ,
• As , , so .
• As , calculate

Step 3: Evaluate .

• Let . Taking log,
• Using and approximation gives
• Thus, Hence, .

Simplifying and applying the condition of continuity:

Step 4: Use differentiability condition at .

• Condition
• Differentiate
• Differentiate using the chain rule.

Complex form of derivative will yield . Evaluating , and substition using continuity condition:

• Simplifying will be .

Step 5: Substituting using options for correctness:

• Correct Option:

Thus the value of is .

We are asked to find the value of the definite integral for the given piecewise function, which is stated to be differentiable over . This requires us to first find the values of the constants and by using the conditions for differentiability.

Concept Used:

1. Continuity: For a function to be differentiable at a point, it must first be continuous at that point. For the given piecewise function, continuity at the boundary point implies that the left-hand limit (LHL) must equal the right-hand limit (RHL).

2. Differentiability: For the function to be differentiable at , the left-hand derivative (LHD) must equal the right-hand derivative (RHD) at that point.

3. Integration of a Piecewise Function: To evaluate the definite integral of a piecewise function over an interval, we must split the integral at the points where the function's definition changes.

where the function definition changes at .

Step-by-Step Solution:

Step 1: Apply the condition of continuity at .

The left-hand limit is:

The right-hand limit is:

For continuity, we must have LHL = RHL:

Step 2: Apply the condition of differentiability at .

First, we find the derivatives for each piece of the function.

For , . The left-hand derivative at is:

For , . The right-hand derivative at is:

For differentiability, we must have LHD = RHD:

Step 3: Solve for the constants and .

We found . Substituting this value into Equation 1:

So, the function is:

Step 4: Evaluate the definite integral .

Since the function definition changes at , we split the integral at this point.

Step 5: Calculate each integral separately.

For the first integral:

For the second integral:

Final Computation & Result:

Add the results of the two integrals to get the final answer.

The value of the integral is 17.

Let's solve the problem to find the maximum and minimum values of the function in the interval . We need to find , where is the maximum value, and is the minimum value.

The first step is to find the critical points of the function by differentiating it with respect to and setting the derivative to zero.

Using the product rule of differentiation, if and , then:

Applying the product rule:

Setting , we solve:

Factor out :

This simplifies to:

Solve each factor:

• →
• →
• →

Evaluate at the critical points and endpoints of the interval :

From the above calculations, we find that and . Therefore, the value of .

Thus, the correct answer is .

To solve the problem, we need to analyze the function given:

Function: where .

Step 1: Analyze Continuity

The function can be rewritten as follows: Analyzing the absolute value, can be expressed as:

• If , then , and hence .
• If , then , and hence .

Thus,

• For , .
• For , .

At , .

Step 2: Check Points of Discontinuity

The function is composed of continuous functions on intervals and . At , the function's left-hand limit and right-hand limit should be checked:

• As , .
• As , .

Since both limits are equal to the value of the function at , the function is continuous at . Thus, .

Step 3: Check Points of Non-Differentiability

The derivatives on either side of should be checked:

• For , differentiates to .
• For , differentiates to .

The derivatives from the left and right at are:

• As , the derivative is .
• As , the derivative is .

Since these derivatives do not match, the function is not differentiable at . Thus, .

Conclusion

The value of . Therefore, the correct answer is .

Solution for the Piecewise Function and Area Enclosed

We are given the function as:

Where and , and the function is differentiable for all . We need to find the area enclosed by the curve and the line , which is expressed as , where and are integers. Finally, we need to find .

Step 1: Finding and for Continuity and Differentiability

- For the function to be continuous and differentiable at , the left-hand and right-hand values of and their derivatives must match. - **Continuity condition:** For to be continuous at , we must have: Simplifying: Thus, we get: - **Differentiability condition:** For to be differentiable at , we must have: At , for differentiability: Thus, we get: - Solving the two equations and , we find: Simplifying: Factoring: Since , we choose . Substituting into , we get:

Step 2: Finding the Area

The total area is the sum of two areas: - **Area for **: The function is . The area under the curve and above is: Solving the integral: - **Area for **: The function is . The area under the curve and above is: Evaluating this integral: As , the integral diverges. However, we calculate the area in the region where the curve is bounded. The final result from calculations gives us:

Step 3: Conclusion

Therefore, the value of is:

We are asked to find the number of 3-digit numbers divisible by 6 but not divisible by 36.

Step 1: Find the total number of 3-digit numbers divisible by 6

A number is divisible by 6 if it is divisible by both 2 and 3. The first three-digit number divisible by 6 is 102, and the last three-digit number divisible by 6 is 996. To find the total number of 3-digit numbers divisible by 6, we use the formula for the number of terms in an arithmetic sequence: Substituting the values: Thus, there are 150 numbers divisible by 6.

Step 2: Find the total number of 3-digit numbers divisible by 36

A number is divisible by 36 if it is divisible by both 4 and 9. The first three-digit number divisible by 36 is 108, and the last three-digit number divisible by 36 is 972. Using the same formula for the number of terms: Thus, there are 25 numbers divisible by 36.

Step 3: Subtract the numbers divisible by 36 from those divisible by 6

The required number of 3-digit numbers that are divisible by 6 but not divisible by 36 is:

Final Answer:

The required number of 3-digit numbers is .

Concept:
The composition can be discontinuous when:

• is discontinuous
• crosses the boundary where the definition of changes

Here, changes definition at .

Step 1: Write the expression for .
For : Thus:
Step 2: Find where .
Thus, changes sign at .

Step 3: Check continuity at .
Left-hand limit: Right-hand value: Thus, is discontinuous at , so is also discontinuous at .

Step 4: Check critical points.
Critical points: At both points, the expression of changes, leading to discontinuities.

Final Answer: