Problem 36 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The function \(f(x)=\left[x^{2}\right]+[-x]^{2}\), where [.] denotes the greatest integer function, is (A) continuous and derivable at \(x=2\) (B) neither continuous nor derivable at \(x=2\) (C) continuous but not derivable at \(x=2\) (D) None of these

Step-by-Step Solution

Answer

(B) neither continuous nor derivable at \(x=2\).

Step 1: Understand the Function Components

The function is composed of two parts: \(f(x) = [x^2] + [-x]^2\). The square brackets denote the greatest integer function, which represents the largest integer less than or equal to the value inside.

Step 2: Evaluate Each Component at x=2

For \([x^2]\), when \(x = 2\), we compute \(2^2 = 4\). Since 4 is already an integer, \([4] = 4\). For \([-x]^2\), when \(x = 2\), \(-x = -2\), and \([-2]^2 = (-2)^2 = 4\). Therefore, \(f(2) = [4] + 4 = 4 + 4 = 8\).

Step 3: Check Continuity at x=2

To determine continuity, check the left-hand limit (\(f(x)\) as \(x\) approaches 2 from the left) and right-hand limit (\(f(x)\) as \(x\) approaches 2 from the right), and compare them to \(f(2)\). If both limits and \(f(2)\) are equal, the function is continuous at \(x = 2\).

Step 4: Compute Limits as x Approaches 2

For \(x\to 2^-, x^2\) approaches 4.99, so \([x^2] = 3\). For \([-x]^2\) as \(x\to 2^-, -x\) approaches just below -2, so \([-x]^2 = 4\). Therefore, the left-hand limit, \(3 + 4 = 7\). For \(x\to 2^+, x^2\) is just above 4, so \([x^2] = 4\). \([-x]^2\) still results in 4, so the right-hand limit is \(4 + 4 = 8\).

Step 5: Determine Continuity

Since the left-hand limit is 7 while the right-hand limit and \(f(2)\) are both 8, \(f(x)\) is not continuous at \(x = 2\).

Step 6: Derivability Analysis

Since \(f(x)\) is not continuous at \(x = 2\), it cannot be derivable at that point. A function must be continuous to be derivable.

Key Concepts

Continuity

Continuity is about how smooth a function behaves at a particular point. If a function is continuous at a point, then there is no abrupt jump or hole in its graph at that point. Here, we examine the function at the point where the variable approaches 2. This involves:

• Evaluating the function value, which is known as the actual point value: here, it is \(f(2) = 8\).
• Calculating the left-hand and right-hand limits. For continuity, these limits, along with the actual value, should all be equal.

To assess continuity at \(x = 2\), the left-hand limit was found to be 7, while the right-hand limit and the actual function value were both 8. Due to this difference, the function is not continuous at \(x = 2\). If these had all matched, \(f(x)\) would be continuous.

Derivability

Derivability is the property of a function that determines whether it has a derivative at a particular point. A derivative represents a function's rate of change at that specific point. One key factor is that a function must be continuous to have a derivative.To check whether a function is derivable:

• Ensuring continuity is crucial. Since our function is not continuous at \(x = 2\), it automatically means it is not derivable at this point either.
• If a function was continuous, we would then look at how the limits of its difference quotient (slope of the tangent) behave as \(x\) approaches the point in question.

In this case, because \(f(x)\) is not continuous at \(x = 2\), it is also not derivable. No smooth tangent line can be drawn at that point, reflecting the lack of a defined slope.

Limits

The concept of limits helps to understand the behavior of functions as they approach specific points. Limits allow us to study the approach of a function toward a value, revealing critical information about continuity and potential jumps or gaps.For the exercise, we consider:

• Left-hand limit: This is the value that \(f(x)\) approaches as \(x\) nears 2 from the left. It was shown to be 7.
• Right-hand limit: This is the value \(f(x)\) approaches as \(x\) approaches 2 from the right. This was calculated as 8.
• Comparing these limits with \(f(2)\) helps in determining continuity. Here, since the left-hand limit differs from the right-hand limit and \(f(2)\), we conclude discontinuity.

Limits offer a powerful toolkit to analyze how functions behave and confirm assumptions on continuity and derivability.