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.