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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

This function has $50$ local maxima

Vedclass · Answer

(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?

Similar Questions

If the function $f(x)=xe^{-x}, x \in R$ attains its maximum value $\beta$ at $x=\alpha$, then $(\alpha, \beta)=$

$N$ characters of information are held on magnetic tape,in batches of $x$ characters each; the batch processing time is $\alpha + \beta x^2$ seconds; $\alpha$ and $\beta$ are constants. The optimal value of $x$ for fast processing is

If the curved surface area of a right circular cylinder inscribed in a sphere of radius $22 \ cm$ is maximum,then the height of the cylinder will be:

Let $f: R \rightarrow R$ be a function defined by $f(x) = (x - 3)^{n_{1}}(x - 5)^{n_{2}}$,where $n_{1}, n_{2} \in N$. Which of the following is $\text{NOT}$ true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module