Problem 79 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The function \(f(x)=\max .\\{(1-x),(1+x), 2\\}\) \(x \in(-\infty, \infty)\), is (A) continuous at all points (B) differentiable at all points (C) differentiable at all points except at \(x=1\) and \(x\) \(=-1 .\) (D) continuous at all points except at \(x=1\) and \(x=-1\), where it is discontinuous.

Step-by-Step Solution

Answer

The function is continuous at all points and non-differentiable at \(x=1\) and \(x=-1\), so (C) is correct.

Step 1: Understanding the Function

The function given is \( f(x) = \max \{ 1-x, 1+x, 2 \} \). This means for each \( x \), we take the maximum value from the set \( \{ 1-x, 1+x, 2 \} \). Essentially, the function will switch between these expressions depending on which one yields the largest result.

Step 2: Finding the Point of Change

To determine where the function might have discontinuities or non-differentiable points, check where these expressions intersect. Solve for intersections: \(1-x = 2\), \(1+x=2\), and \(1-x=1+x\). The solutions are \(x=-1\) and \(x=1\), meaning potential points of change at these \( x \) values.

Step 3: Checking Continuity

We check every possible expression around \(x=-1\) and \(x=1\) for continuity. The function will be continuous if at \(x=1\) and \(x=-1\), the value from each side matches the value of the function at these points. Evaluate limits from left and right at \(x=1\) and \( x=-1\), and they match the function value, confirming continuity.

Step 4: Checking Differentiability

\(f(x)\) is differentiable if it is continuous and smooth. At points where it switches expressions (\(x=1\) and \(x=-1\)), examine the derivatives from the left and right. There is a cusp or corner because different potential derivatives \(( -1,1 )\) result in a nondifferentiable point due to mismatch between the left-hand and right-hand derivatives.

Step 5: Conclusion

From the analysis, \(f(x)\) is continuous across the real number line since it does not have jump or removable discontinuities anywhere. However, \( f(x) \) is non-differentiable at \( x=1 \) and \( x=-1 \) due to changes in maximum function expressions and non-matching derivatives.

Key Concepts

Continuity

A function is considered continuous if there are no interruptions, jumps, or breaks in its graph. In a more formal definition, continuity at a point means that the limit of the function as it approaches the point from both directions equals the function's value at that point. For the function given, \( f(x) = \max \{1-x, 1+x, 2\} \), it's crucial to check continuity at the critical points identified, which are \( x = 1 \) and \( x = -1 \).
When we approach these points from the left and right, the limits need to be equal to ensure continuity. We substitute values just less than and greater than \( x = 1 \) and \( x = -1 \), confirming the function value remains consistent as the limits from the left side equal those from the right side.
This confirms that no jumps or holes are present in the graph at these points. Therefore, because these conditions are satisfied across the entire domain, the function is continuous everywhere.

Differentiability

Differentiability of a function is closely related to its smoothness, meaning there should be no sharp turns or cusps in its graph. However, even if a function is continuous at a certain point, it may not be differentiable there. For the function \( f(x) = \max \{1-x, 1+x, 2\} \), differentiability is investigated, especially near points \( x = 1 \) and \( x = -1 \) where expressions switch.
At \( x = 1 \), the left-hand derivative is derived from \( 1-x \), while the right-hand derivative comes from \( 1+x \). Calculating both derivatives gives us values of \(-1\) and \(1\). Due to these differing derivatives, the function is not smooth and thus non-differentiable at this point. The same procedure applies to \( x = -1 \), with similar results.
While the function is continuous, this analysis reveals that it isn't differentiable at \( x = 1 \) and \( x = -1 \) because of these abrupt changes or corners in the graph.

Real Analysis

Real analysis is a field in mathematics focusing on real numbers and real-valued functions. Key components include understanding concepts such as limits, continuity, and differentiability. It's a foundational subject that allows us to analyze functions with precision and rigor.
For example, in the given function \( f(x) = \max \{1-x, 1+x, 2\} \), real analysis provides the frameworks to assess the continuity and differentiability of the function. It involves examining changes in the function’s expressions at points where these switches occur, like at \( x = 1 \) and \( x = -1 \).
These points are where potential discontinuities or non-differentiable "corners" might arise. Real analysis equips students with the tools to rigorously determine these properties, ensuring proper understanding and application in a variety of contexts, such as mathematical proofs or applied problems.