Problem 120 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\min\) \(\\{x+1,|x|+1\\} .\) Then which of the following is true? \([2007]\) (A) \(f(x) \geq 1\) for all \(x \in \mathrm{R}\) (B) \(f(x)\) is not differentiable at \(x=1\) (C) \(f(x)\) is differentiable everywhere (D) \(f(x)\) is not differentiable at \(x=0\)

Step-by-Step Solution

Answer

(A) is true, but (D) is the correct description of differentiability.

Step 1: Understand the Function

The function is defined as \( f(x) = \min \{ x+1, |x| + 1 \} \). This means for each \( x \), the function takes the smaller of \( x+1 \) and \( |x| + 1 \). We will analyze these separately to understand how the function behaves.

Step 2: Evaluate for positivity

Both expressions \( x+1 \) and \( |x|+1 \) are always greater than or equal to 1 for all real \( x \). Specifically, \( |x| \geq 0 \) so \( |x|+1 \geq 1 \). Thus, \( f(x) \geq 1 \) follows.

Step 3: Check differentiability at 0

Evaluate the sub-functions at \( x=0 \): \( x+1 = 1 \) and \( |x|+1 = 1 \). Since both are equal, the minimum is well defined and continuous. Check differentiability of these sub-functions at \( x=0 \): \( x+1 \) with derivative 1, \( |x|+1 \) with derivative \( -1 \to 1 \). Hence, \( f(x) \) is not differentiable at \( x = 0 \).

Step 4: Examine differentiability at 1

At \( x=1 \), \( f(x) \) evaluates \( x+1 = 2 \) and \( |x|+1 = 2 \), thus continuous and checks differentiability follows from \( x=1 \) derivatives as both are \( 1 \). Hence differentiable at \( x=1 \).

Step 5: Shortlist Correct Statements

Given analysis, \( f(x) \geq 1 \) is true (A), and \( f(x) \) is not differentiable at \( x=0 \) (D). It is differentiable everywhere else; hence \( f(x) \) is differentiable at \( x=1 \). Thus (D) holds.

Key Concepts

Real-valued functions

Real-valued functions are a fundamental concept in mathematics where both the input and the output of the function are real numbers. This means that when you plug in a real number into the function, you also get a real number out. These functions are crucial because they form the basis for many mathematical models in science and engineering.

For example, the function given in the exercise, \( f(x) = \min \{ x+1, |x| + 1 \} \), is a real-valued function. The domain here is all real numbers, \( \mathbb{R} \), and for any real number that you input, the function will output another real number. The range of this particular function also lies within the real numbers.

In practical terms, considering real-valued functions is beneficial because it allows us to apply continuous mathematics, such as calculus, and analyze change, limits, and behavior of these functions in a very rich and detailed manner. Analyzing differentiability, for example, is crucial in understanding how smoothly a function behaves, which involves studying whether the function's derivatives exist at various points.

Piecewise functions

Piecewise functions are functions that have different expressions or rules for different intervals of the domain. They are essential in mathematics because they model situations where a function behaves differently across various pieces of its domain, akin to turning a light switch on and off.

In the given exercise, the function \( f(x) = \min \{ x+1, |x| + 1 \} \) operates piecewise since it chooses between two different expressions \( x+1 \) and \( |x|+1 \). For different real numbers \( x \), it will output the smaller result between these two options. This can be visualized as:

• If the linear part \( x+1 \) is smaller, the function follows \( x+1 \).
• If the absolute part \( |x|+1 \) is smaller, it follows \( |x|+1 \).

When analyzing piecewise functions, an essential aspect is checking for continuity and differentiability around the points where the expression changes, known as boundary points or transition points. By doing so, you can understand how the function behaves at these critical points, as was done in steps of the solution at \( x = 0 \) and \( x = 1 \).

Min-max functions

Min-max functions, also known as extremum functions, involve choosing the smallest or largest value from a given set of options. They are highly effective in modeling decisions and optimizations in various fields, such as economics, operations research, and computer science.

In the context of the function \( f(x) = \min \{ x+1, |x| + 1 \} \), the function is based on taking the minimum value from the set \( \{ x+1, |x|+1 \} \). This means at any given point \( x \), the function evaluates both expressions and selects the lesser of the two values to output.

Understanding min-max functions requires an understanding of how these choices change as parameters change. For instance, in this exercise, exploring where \( x+1 \) and \( |x|+1 \) are equal or which is smaller helps in determining points of non-differentiability or where different rules apply. Consequently, analysis of min-max functions often involves examining the behavior around these critical points and understanding the choice transformation as \( x \) changes, offering insight into the function’s structural design and operational outcomes.