Problem 25 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=(-1)^{\left[x^{2}\right]}\), where \([.]\) denotes the greatest integer function, then (A) \(f(x)\) is discontinuous for \(x=n^{1 / 3}\), where \(n \in I\) (B) \(f(3 / 2)=1\) (C) \(f^{\prime}(x)=0\) for \(-1

Step-by-Step Solution

Answer

Options (A), (B), and (C) are correct.

Step 1: Understanding the Function

The function is given as \( f(x) = (-1)^{[x^2]} \). The symbol \([.]\) represents the greatest integer function, also known as the floor function, which returns the greatest integer less than or equal to the given number.

Step 2: Analyzing Discontinuity

To check where \( f(x) \) is discontinuous, we need to determine if \( (-1)^{[x^2]} \) changes value abruptly for small changes in \( x \). Notice that \( [x^2] \) changes value at integer squares, suggesting potential discontinuity at cube roots of integers (i.e., \( x = n^{1/3} \) where \( n \) is an integer). Therefore, option (A) is likely true.

Step 3: Evaluate for Specific Value

Calculate \( f(3/2) \): \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} = 2.25 \). Thus, \( [2.25] = 2 \). Therefore, \( f\left(\frac{3}{2}\right) = (-1)^2 = 1 \). Thus, option (B) is correct.

Step 4: Check Derivative Condition

For \( -1 < x < 1 \), \( x^2 < 1 \), so \( [x^2] = 0 \) for all such \( x \). Therefore, \( f(x) = (-1)^0 = 1 \), which is constant. This would imply \( f'(x) = 0 \) in this range, making option (C) correct.

Key Concepts

greatest integer function

The greatest integer function, denoted by \([.]\), is often referred to as the floor function. It is a mathematical function that takes any real number as input and gives the greatest integer that is less than or equal to the input number as output. For example, the greatest integer function of 2.9 is 2, and the greatest integer function of -1.2 is -2. It essentially "rounds down" a number to the nearest integer.

• Useful in algorithms and computer science for integer division.
• Provides a piecewise constant function, which is often discontinuous.

In the context of the given exercise, the greatest integer function is applied to \(x^2\), which can lead to changes in the function value at specific points, contributing to discontinuity. Understanding how \[x^2\] changes is key in predicting the points where the function \(f(x) = (-1)^{[x^2]}\) may behave unexpectedly.

discontinuity analysis

Discontinuity in functions occurs when there is a jump, break, or hole in the graph of the function. For the function \(f(x) = (-1)^{[x^2]}\), discontinuity can occur where \[x^2\] results in a different greatest integer for infinitesimally small changes in \(x\). The excursion from one integer value of \[x^2\] to the next introduces points where the function may abruptly change its value:

• Discontinuity in greatest integer functions: typically occurs at integer boundaries, where the output of the function jumps from one integer to the next.
• Critical points for \(f(x)\) are therefore where \(x^2\) crosses an integer boundary, notably at \(x = n^{1/3}\), where the cube of \(x\) results in an integer.

Analyzing these points helps in determining discontinuity, and ultimately confirms whether or not a function remains reliable and consistent across stretches of its domain.

function derivatives

Derivatives represent the rate of change of a function. For constant functions, their derivative is straightforward — it is zero, as there is no change in function value with respect to change in the independent variable.

• In the exercise, for the interval \(-1 < x < 1\), the function \(f(x)\) simplifies to a constant \(1\) because \([x^2] = 0\).
• With \(f(x) = 1\) being constant over that interval, \(f'(x) = 0\) demonstrates no change or slope.

This concept reinforces how derivatives explain the behavior of a function over specific intervals, clarifying that while \(f(x)\) behaves constantly within certain bounds, variability elsewhere is possible.

mathematical problem solving

Mathematical problem solving is a systematic process of applying mathematical methods to analyze complex scenarios and arrive at a solution. This approach demands:

• Understanding the problem: Fully grasping all elements described, such as the function definition \(f(x) = (-1)^{[x^2]}\).
• Analyzing components: Breaking down the function into simpler parts (e.g., evaluating \[x^2\]).
• Recognizing patterns and behavior: Such as identifying where discontinuities might appear.
• Justifying outcomes: Explaining why certain intervals or results meet the problem's conditions (e.g., why \(f'(x) = 0\) for \(-1 < x < 1\)).

Through a structured approach, problem solving not only solves math problems but also develops a systematic way of thinking that can be applied broadly across challenges.