Problem 47 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(f(x)=\left\\{\begin{array}{cc}\max \left\\{|x|, x^{2}\right\\} & |x| \leq 2 \\ 8-2|x|, & 2<|x| \leq 4\end{array}\right.\) Let \(\mathrm{S}\) be the set of points in the interval \((-4,4)\) at which \(f\) is not differentiable. Then S: \(\quad\) (a) is an empty set (b) equals \(\\{-2,-1,0,1,2\\}\) (c) equals \(\\{-2,-1,1,2\\}\) (d) equals \(\\{-2,2\\}\)

Step-by-Step Solution

Answer

S equals \(\{-2,-1,1,2\}\).

Step 1: Understand the Piecewise Function

The function \( f(x) \) is given piecewise with two different expressions depending on the value of \( |x| \). For \( |x| \leq 2 \), \( f(x) = \max(|x|, x^2) \). For \( 2 < |x| \leq 4 \), \( f(x) = 8 - 2|x| \). We need to analyze each segment separately to identify points where the function is not differentiable.

Step 2: Analyze the Interval \\((-2, 2)\\)

For \( |x| \leq 2 \), calculate \( \max(|x|, x^2) \). For \( -2 \leq x < -1 \), \( |x| = -x \) and \( x^2 \geq |x| \); hence \( f(x) = x^2 \). For \( -1 \leq x < 0 \) and \( 0 < x \leq 1 \), \( x^2 < |x| \) thus \( f(x) = |x| \). From \( 1 < x \leq 2 \), \( x^2 > |x| \) so \( f(x) = x^2 \). Check points \( x=-2, -1, 0, 1, 2 \) for non-differentiability.

Step 3: Analyze the Interval \\((2, 4)\\) and \\((-4, -2)\\)

For \( 2 < |x| \leq 4 \), \( f(x) = 8 - 2|x| \). The expression should be both continuous and differentiable as it is a linear function. Double check continuity at transition points \( x = -2 \) and \( x = 2 \) where function expressions change.

Step 4: Check Continuity and Differentiability at Potential Points

At \( x = -2 \), \( f(x) \) transitions from \( x^2 \) to \( 8 - 2|x| \) and these have different derivatives. Similarly, \( f(x) \) transitions at \( x = 0 \) and at \( x = 2 \) from \( \max(|x|, x^2) \) segments. Calculate left-hand and right-hand derivatives at these points to confirm non-differentiability points.

Step 5: List Non-differentiable Points

After verifying derivatives on each identified point, \( f(x) \) is not differentiable at \( x = -2, -1, 1, \) and \( 2 \) due to the change in expressions or kinks at these specific points. Confirm continuity where function smoothly transitions between segments.

Key Concepts

Piecewise Function

A piecewise function is a function that has different definitions for different intervals of the input variable, usually separated by certain points or values. In this exercise, the function \( f(x) \) is defined differently for \( |x| \leq 2 \) and \( 2 < |x| \leq 4 \).

These types of functions are common in mathematical analysis when the behavior of the function changes at certain points or over specific intervals. It's important to understand each segment of a piecewise function individually, as each piece has its own rules and often its own graph shape. In our example:

• For \( |x| \leq 2 \), the function \( f(x) = \max(|x|, x^2) \).
• For \( 2 < |x| \leq 4 \), the function \( f(x) = 8 - 2|x| \).

By analyzing each piece separately, you can determine the overall behavior of the function, including where it might not be continuous or differentiable.

Continuity

Continuity in a function means it has no breaks, jumps, or holes at any given point in its domain. For piecewise functions, continuity must be checked at points where the function's definition changes, known as transition points.

In this problem, the transition points are at \( x = -2 \) and \( x = 2 \). To check continuity at these points, we need to ensure that the limit approaching from the left equals the limit approaching from the right, as well as the function's value at that point.

• At \( x = -2 \), consider both limits and function values from the definitions of \( x^2 \) and \( 8 - 2|x| \).
• At \( x = 2 \), reevaluate continuity between the function portions \( x^2 \) and \( 8 - 2|x| \).

Simply examining these limits can often reveal disruptions in continuity, meaning the function doesn't smoothly transition from one piece to another.

Derivative

A derivative measures how a function changes as the input changes and is represented by the slope of the tangent line at any given point on the function's graph. For piecewise functions, differentiability can be tricky, particularly at transition or corner points.

In this exercise, to find where \( f(x) \) is non-differentiable, you must:

• Check points where the function transitions from one piece to another.
• Compute the derivatives from the left and the right of these points to see if they match. If they differ, the function isn't differentiable there.

Points like \( x = -2, -1, 1, \) and \( x = 2 \) are considered for their derivative evaluations. At these points, either the function changes its form involving different rules or abrupt directional changes, indicating non-differentiability.

Mathematical Analysis

Mathematical analysis provides tools and methods for rigorously examining mathematical functions, focusing largely on the concepts of limits, continuity, and differentiability.

In problems involving functions like the one in this exercise, mathematical analysis helps:

• Determine where functions transition between continuous and discontinuous states.
• Identify points of non-differentiability which are critical due to changes like corners or cusps.
• Ensure a logical and systematic approach when dealing with complex or layered function expressions.

Through careful analysis of the behavior across different function segments, more comprehensive conclusions about the function's properties, such as smoothness and rate of change, can be drawn.