Problem 26 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)=\left[\frac{1}{\sqrt{2}}(\cos x+\sin x)\right], 0

Step-by-Step Solution

Answer

The number of discontinuities is 2.

Step 1: Simplify the Function

Consider the function \( f(x) = \left[ \frac{1}{\sqrt{2}} (\cos x + \sin x) \right] \). We begin by analyzing the part inside the brackets: \( \frac{1}{\sqrt{2}}(\cos x + \sin x) \). Using the identity \( \cos x + \sin x = \sqrt{2} \sin(x + \frac{\pi}{4}) \), we rewrite it as \( \sin(x + \frac{\pi}{4}) \). Thus, our function becomes \( f(x) = [\sin(x + \frac{\pi}{4})] \).

Step 2: Analyze the Range of the Sine Function

The function inside the brackets, \( \sin(x + \frac{\pi}{4}) \), varies between -1 and 1 as \( x \) goes from \( 0 \) to \( 2\pi \). The values of \( x \) where \( \sin(x + \frac{\pi}{4}) \) crosses integer boundaries will create discontinuities in \( f(x) \) due to the greatest integer function applied.

Step 3: Identify Points Where Discontinuities Occur

Discontinuities occur when \( \sin(x + \frac{\pi}{4}) = 0 \). Solving for \( x \), we get \( x + \frac{\pi}{4} = n\pi \) where \( n \) is an integer. Solving this equation for \( x \) within the interval \( (0, 2\pi) \), we find the values \( x = \frac{3\pi}{4}, \frac{7\pi}{4} \).

Step 4: Count the Discontinuities

Each value of \( x \) determined in the previous step represents a point of discontinuity of the function. Therefore, \( f(x) \) has discontinuities at \( x = \frac{3\pi}{4}, \frac{7\pi}{4} \). In total, there are 2 points of discontinuity within the interval \( (0, 2\pi) \).

Key Concepts

Discontinuity

In mathematical terms, a discontinuity is a point where a function is not continuous. For the given function, which is defined using the greatest integer function \(f(x) = \left[\sin(x + \frac{\pi}{4})\right]\), discontinuities arise where there are sudden jumps in the value. These jumps occur because the greatest integer function, also known as the floor function, changes its output integer when its input crosses an actual integer value.

• This function is discontinuous where \(\sin(x + \frac{\pi}{4})\) equals zero, as these points mark the boundary of integer and non-integer results.
• The function only has discontinuities where the sine function exactly hits an integer, owing to the nature of the greatest integer stepping function.

As analyzed in the solution, these precise points within the interval \(0 < x < 2\pi \) were determined to be at \( \frac{3\pi}{4}, \, \frac{7\pi}{4}\).The function behaves differently before and after crossing each discontinuity, leading to a total of two points where the function is not continuous within the specified interval.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. In this exercise, we employ the identity \(\cos x + \sin x = \sqrt{2} \sin(x + \frac{\pi}{4})\).

• This identity is instrumental in simplifying the expression used within the given function to \(\sin(x + \frac{\pi}{4})\).
• Such identities are useful because they transform complex trigonometric expressions into a simpler form, making it easier to identify key points, like zeros and amplitude changes.

By utilizing this identity, the expression inside the greatest integer function becomes more manageable, allowing us to efficiently find where discontinuities occur. Sine, known for its wave-like oscillations between -1 and 1, clearly highlights the transitions worth noting when paired with the floor function. This identity also assures that the periodic nature of sine can be appropriately analyzed over the interval \(0 < x < 2\pi\), as converting the function helps reveal crucial intersections with integer values.

Sine Function

The sine function, \(\sin(x)\), is a fundamental trigonometric function known for its smooth, periodic oscillations. In the context of this exercise, it plays a central role, as the behavior of \(\sin(x + \frac{\pi}{4})\) determines the nature of the given function \(f(x)\).

• This modified sine function inside our floor operation shifts the basic sine wave horizontally by \(\frac{\pi}{4}\).
• Its periodic nature means it will complete full oscillations from -1 to 1 and back as \(x\) goes from 0 to \(2\pi\).
• The key to identifying discontinuities lies in these oscillations, where the sine function crosses zero or reaches integer boundaries.

This sine function's oscillation between -1 and 1 creates fluctuations that interact with the discontinuous nature of the floor function. When \(\sin(x + \frac{\pi}{4})\) crosses integer thresholds, it defines exactly where discontinuities will occur, revealing the precise conditions for non-continuity. Understanding the sine function's behavior is critical to discern pivotal points of change or jumps in the function's graph.