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 polynomial of degree 101, with roots at \( x = 0, 1, 2, \dots, 100 \). 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 \( f(x) \) is given by the formula:

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

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

• The first turning point (starting from \( x = 0 \)) is a local maximum.
• This 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:

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

6. The remaining 49 turning points are local minima.