Question

In the following \([x]\) denotes the greatest integer less than or equal to \(x\). Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the \(4 \times 4\) matrix given in the ORS. Column I (A) \(x|x|\) (B) \(\sqrt{|x|}\) (C) \(x+[x]\). (D) \(|x-1|+|x+1|\) Column II (p) continuous in \((-1,1)\) (q) differentiable in \((-1,1)\) (r) strictly increasing in \((-1,1)\) (s) not differentiable at least at one point in \((-1,1)\)

Short answer

(A) matches with (p), (r), (s); (B) matches with (p), (s); (C) does not match with any of the properties; (D) matches with (p), (s) but not with (q).

Step-by-step solution

Analyzing Function A

Function A is given by \(f(x) = x|x|\). This function is the product of \(x\) and \(|x|\). Since \(|x|\) is not differentiable at \(x = 0\), neither is \(f(x)\). However, it is continuous everywhere and is strictly increasing in \((-1,1)\) as both factors \(x\) and \(|x|\) are increasing.

Analyzing Function B

Function B is given by \(f(x) = \sqrt{|x|}\). This function is continuous everywhere as the square root function and the absolute value function are both continuous. However, it’s not differentiable at \(x = 0\) because the absolute value function has a cusp at that point. It's also not strictly increasing, as it stays constant at 0 around \(x = 0\).

Analyzing Function C

Function C is given by \(f(x) = x + [x]\). The greatest integer function \([x]\) has a jump discontinuity at every integer value, which means it is not continuous, not differentiable at these points, and not strictly increasing due to the constant value in the intervals between jumps.

Analyzing Function D

Function D is given by \(f(x) = |x-1| + |x+1|\). This function is formed by the sum of two absolute value functions. It is continuous everywhere but has corners at \(x = 1\) and \(x = -1\), thus not differentiable at these points. However, the function is strictly increasing in \((-1,1)\) as the slope of the graph is positive within that interval.

Matching with Properties

Now we match: (A) matches with (p) because it is continuous, and (r) since it's strictly increasing in the interval. It also matches with (s) because it's not differentiable at 0.(B) matches with (p) for continuity and (s) for not being differentiable at 0.(C) does not match with any of the properties as it's not continuous or differentiable at integer points, nor is it strictly increasing.(D) matches with (p) for continuity and (s) for not being differentiable at at least one point within (-1,1), but it does not match with (q) because of the non-differentiability at the points.

Key concepts

Continuity and Differentiability

Understanding the concepts of continuity and differentiability is crucial for students studying JEE Advanced Mathematics. A function is said to be continuous at a point if there is no interruption in the graph of the function at that point. This essentially means that you could draw the function at this point without lifting the pencil from your paper. Mathematically speaking, a function f(x) is continuous at a point a if the following three conditions are met:

• The function f(x) is defined at a.
• The limit of f(x) as x approaches a exists.
• The limit of f(x) as x approaches a is equal to f(a).

On the other hand, a function is differentiable at a point if it has a well-defined tangent at that point, which implies a definite slope or rate of change. In other words, if you can find the derivative of the function at that point, the function is differentiable there. A function that is differentiable at a point is necessarily continuous at that point, but the converse is not always true.

Let's apply this knowledge. Consider a function with absolute value, like the one in Function D from the exercise, \(f(x) = |x-1| + |x+1|\). While this function is continuous everywhere, it has 'corners' at \(x = 1\) and \(x = -1\), indicating points where the slope changes abruptly. The function is therefore not differentiable at these points, even though it is continuous.

Absolute Value Functions

Absolute value functions, often represented as \(|x|\), measure the distance of a number from zero on the number line without considering the direction. These functions are pivotal in solving many algebraic problems and play a significant role in JEE Advanced Mathematics.

Graphically, an absolute value function looks like a 'V', where the point at the vertex of the 'V' represents the origin of the absolute distance. One of the distinct properties of an absolute value function is that it is always non-negative. It is also continuous everywhere because there is no break in the graph. However, absolute value functions are notoriously known for their non-differentiability at the point where they switch from one arm of the 'V' to the other - typically at the point \(x = 0\) as seen in Function B, \(f(x) = \sqrt{|x|}\), from the exercise. This non-differentiability arises because the direction of the function changes too abruptly, creating a sharp 'cusp' at that point.

Greatest Integer Function

Also known as the 'floor function', the greatest integer function assigns to any real number \(x\) the largest integer less than or equal to \(x\), denoted as \(\left[x\right]\). This function is a step function because it jumps from one integer to the next. It is integral to JEE Advanced Mathematics for handling piecewise-defined functions and solving many problems involving discretization.

The graph of the greatest integer function comprises a series of horizontal steps, which means that this function is not continuous at integer values where these steps occur. As a result, it also fails to be differentiable at these jump discontinuities. In the given exercise, Function C, \(f(x) = x + [x]\), exhibits this property. For intervals that do not contain integers, the function may appear continuous, but since it is not continuous at the integers themselves, we don't consider it continuous over intervals that include these points.