Consider the function f(x) = x(x - 1)(x - 2) | vedclass

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(B) The function is $f(x) = x(x - 1)(x - 2) \dots (x - 100)$.This is a polynomial of degree $101$.The roots of the function are $x = 0, 1, 2, \dots, 1

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This function has $50$ local maxima

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(B) The function is $f(x) = x(x - 1)(x - 2) \dots (x - 100)$.This is a polynomial of degree $101$.The roots of the function are $x = 0, 1, 2, \dots, 100$.Since the leading coefficient is positive and the degree is odd,the graph starts from $-\infty$ and goes to $+\infty$.Between any two consecutive roots $x = k$ and $x = k+1$,there must be at least one local extremum (by Rolle's Theorem).There are $100$ intervals of the form $(k, k+1)$ for $k = 0, 1, \dots, 99$.In each interval,there is exactly one local extremum.Since the function is positive in $(0, 1)$,negative in $(1, 2)$,positive in $(2, 3)$,and so on,the local extrema alternate between local maxima and local minima.The intervals are $(0, 1), (1, 2), (2, 3), \dots, (99, 100)$.Local maxima occur in intervals $(0, 1), (2, 3), \dots, (98, 99)$.The number of such intervals is $50$ (i.e.,$k = 0, 2, \dots, 98$).Thus,there are $50$ local maxima.

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

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