Question

Consider the function
\[ f(x) = x(x - 1)(x - 2) \cdots (x - 100). \]
Which one of the following is correct?

• A. This function has 100 local maxima.
• B. This function has 50 local maxima.
• C. This function has 51 local maxima.
• D. Local minima do not exist for this function.

Hint:

For polynomials, local maxima and minima alternate between consecutive roots. Use symmetry and degree properties to count them efficiently.

Answer 1

The Correct Option is B

1. The function \( f(x) \) is a 101-degree polynomial. Its roots are at \( x = 0, 1, 2, \dots, 100 \), dividing the real line into 100 intervals.

2. The polynomial changes sign between each pair of consecutive roots, indicating turning points (local extrema) in each interval.

3. The number of turning points of \( f(x) \) is:

\[ \text{Number of turning points} = \text{Degree of the polynomial} - 1 = 101 - 1 = 100. \]

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

• The first turning point (starting from \( x = 0 \)) is a local maximum.
• This pattern continues for the other 99 turning points.

5. Because every other turning point is a local maximum, the total number of local maxima is:

\[ \text{Total turning points} + 1 \div 2 = \frac{100 + 1}{2} = 50. \]

6. The remaining 49 turning points are local minima.