Limits

Step 1: The given function is:

Step 2: Find the derivative of to determine if it is increasing or decreasing:

Step 3: The derivative depends on the value of . For large values of , dominates, making , which indicates that is increasing for large .

Step 4: Since is not always positive or negative, the function is neither always increasing nor decreasing.

Step 1: The length is the distance from the fixed point to the point on the curve. We are told that is either a maximum or a minimum.

Step 2: For the length to be at a maximum or minimum, the line connecting and must be perpendicular to the tangent at .

Step 3: The condition for the minimum or maximum distance is that the vector is perpendicular to the tangent line at .

Answer: (A) is perpendicular to the tangent at

1. The sum is:

.

Factor out :

.

2. The term inside the brackets is the -th power sum:

3. Substitute:

4. Divide by :

5. Taking the limit as :

Thus, the correct answer is .

1. The curve is given by:

At , the curve crosses the -axis at .

2. The straight line is given by:

3. Check the intersection point of and :

• At , for both and .
• The line touches the curve at the point where it crosses the -axis.

4. For the -axis:

• The curve approaches as , but it never touches the -axis.

Thus, touches at the point where crosses the -axis.

Step 1: The function is defined as the absolute value of . To analyze the function, we consider two cases for :

This piecewise function describes a parabola that is reflected along the x-axis when and a parabola opening upwards for .

Step 2: Local minima occur where the function reaches its lowest value. First, notice that when because:

At and , the function transitions from decreasing to increasing, indicating that these are points of local minima.

Step 3: Local maxima occur where the function reaches its highest value within a given interval. Notice that the function reaches a local maximum at , because:

The function decreases on the interval and then increases after , so is a local maximum.

Step 4: Therefore, the function has local minima at and a local maximum at .

The determinant of f(x) is:

Step 1: Expand the determinant.

Expand along the first row:

Simplify each minor determinant:

Thus:

Step 2: Simplify .

Divide by :

Simplify each term:

1. First term:
2. Second term:
3. Third term:

Thus:

Step 3: Take the limit as .

Using standard limits:

substitute :

1. First term:
2. Second term:
3. Third term:

Combine terms:

Conclusion: The value of is:

1. The function is a polynomial of degree 101, with roots at . These roots divide the real line into 100 intervals.

2. Between each pair of consecutive roots, the polynomial changes sign. This implies that there are turning points (local extrema) in each interval.

3. The total number of turning points of is given by the formula:

4. Turning points alternate between local maxima and local minima:

• The first turning point (starting from ) is a local maximum.
• This alternation continues across the remaining 99 turning points.

5. Since the first and every alternate turning point is a local maximum, the total number of local maxima is:

6. The remaining 49 turning points are local minima.