Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=\cos 2 \pi x+x-[x]\), where \([-]\) denotes the greatest integer function. Then the number of points in \([0,10]\) at which \(f(x)\) assumes its local maximum value, is (A) 10 (B) 9 (C) 0 (D) infinite

Step-by-Step Solution

Answer

The number of local maximum points is 10.

Step 1: Simplify Function Components

Consider the function \( f(x) = \cos(2\pi x) + x - [x] \). The components to focus on are \( \cos(2\pi x) \) and \( x - [x] \). Notice that \( x - [x] \) represents the fractional part of \( x \), denoted as \( \{x\} \). This means the function can be rewritten as \( f(x) = \cos(2\pi x) + \{x\} \).

Step 2: Analyze Function Behavior

The function \( f(x) = \cos(2\pi x) + \{x\} \) needs to be analyzed in the interval \([n, n+1)\) where \( n \) is an integer and \( \{x\} \) ranges from 0 to 1. Within this interval, since \( \{x\} \) increases from 0 to just below 1 and \( \cos(2\pi x) \) is a periodic function oscillating between -1 and 1, \( f(x) \) reaches its maximum value when \( \cos(2\pi x) \) is at its peak, which is 1.

Step 3: Determine Points of Local Maximum

For \( \cos(2\pi x) = 1 \), \( x \) must be an integer because \( \cos(2\pi n) = 1 \) for integer \( n \). At these points \( f(x) = 1 + \{n\} = 1 \) and since \( \{n\} = 0 \) at integers, the function achieves its local maximum value at these integer points.

Step 4: Count Points in Given Interval

Within the interval \([0, 10]\), the integers are 0, 1, 2, ..., 10. There are 11 points at which \( f(x) \) achieves its maximum value. However, the maximum is not considered 'local' at both ends of a boundary in grouped conditions, excluding the point at 10, because at each boundary adjacent points must exist with lower values. Thus, there are 10 such local maximum points remaining, as we exclude \( x = 10 \).

Key Concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, is denoted by \([x]\). It represents the greatest integer that is less than or equal to a given number \(x\). To understand this better, imagine you have a number line and you're "rounding down" any real number to the nearest whole number on the line.

• For example, for \(x = 3.7\), \([3.7] = 3\).
• If \(x = -2.3\), then \([-2.3] = -3\).
• And \(x = 5\), so \([5] = 5\).

The greatest integer function plays a crucial role in mathematical functions involving integers, like the function in this problem where \([x]\) is used to extract the integer part of \(x\), effectively separating whole numbers and their fractional parts. This becomes particularly relevant when dealing with mixed functions such as \(f(x) = \cos(2\pi x) + x - [x]\).

Fractional Part

The fractional part of a number focuses on what's left once you've extracted the integer part. It is represented by \(\{x\} = x - [x]\). This means that for any real number \(x\), the fractional part, \({x}\), is the "leftover" value when the greatest integer is subtracted from the number itself.
Imagine you are breaking down the number into integer and decimal parts:

• If \(x = 4.25\), then \([x] = 4\) and \({x} = 0.25\).
• For \(x = 7.5\), \([x] = 7\) and \({x} = 0.5\).
• When \(x = 3\), \([x] = 3\) and \({x} = 0\), since whole numbers have no fractional part.

In the context of the function \( f(x) = \cos(2\pi x) + \{x\}\), the fractional part \(\{x\}\) adds a linear component that increases from 0 to just below 1 as \(x\) progresses from one integer to the next, impacting the overall behavior of the function.

Periodic Function

A periodic function is one that repeats its values at regular intervals or periods. A classic example is the trigonometric function \(\cos(2\pi x)\), which has a period of 1. This means that every increase of 1 unit in \(x\) results in the function value repeating itself.

• The \(\cos\) function oscillates between -1 and 1, peaking at 1 and reaching a minimum of -1.
• Its periodic nature means that the function \(\cos(2\pi x)\) is always centered around zero and cycles its values across each interval [\(n, n+1\)] where \(n\) is an integer.
• Within each period, the function value returns to 1 at integer points, such as \(n\), causing these points to potentially be locations for local maximums when combined with other function components.

In the original problem, \(\cos(2\pi x)\)'s periodic behavior ensures that it contributes predictably to the function value, creating certain points where \(f(x)\) achieves maximum values when considered along with the fractional part.