Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=\left\\{\begin{array}{ll}|x|, & 0<|x| \leq 2 \\ 1, & x=0\end{array}\right.\). Then, at \(x=0, f\) has (A) a local maximum (B) no local maximum (C) a local minimum (D) no extremum

Step-by-Step Solution

Answer

(A) a local maximum

Step 1: Understand the Function

The function is defined piecewise which means it behaves differently at different intervals. It uses the absolute value of \(x\) for \(0 < |x| \leq 2\) and a constant value of 1 at \(x = 0\). This tells us how \(f(x)\) is structured around \(x = 0\).

Step 2: Analyze Around x=0

For values very close to \(x = 0\), but not equal to zero, \(f(x) = |x|\). By evaluating the function as \(x\) approaches 0 from both the positive and negative directions, we get \(f(x) = |x| \to 0\). At \(x=0\), the function takes the value \(f(x)=1\).

Step 3: Determine Local Extrema

A local maximum or minimum at \(x=0\) would involve \(f(x)\) being larger or smaller than \(f(x)\) for values closely surrounding \(x=0\). Specifically, for a local maximum at \(x=0\), we would need \(f(0) \geq f(x)\) for values near 0, but since \(f(x)\to 0\) as \(x\to 0\), \(f(0)=1\) is indeed larger than \(f(x)\) near \(x=0\). Thus \(x=0\) is a local maximum point.

Key Concepts

Local Maximum

In mathematical terms, a local maximum occurs at a point if the function value at that point is greater than or equal to values at points in a nearby interval. When analyzing piecewise functions, identifying a local maximum is crucial because it determines where a function reaches a peak within a localized region.

For the given piecewise function, check values around a specific point, such as at the origin in this function. At \(x = 0\), the function \(f(x)\) takes the value 1. If we compare this to values just surrounding \(x = 0\), the function \(f(x) = |x|\) moves towards zero as \(x\) approaches zero from both sides. Hence, \(f(0) = 1\) indeed forms a peak relative to these surrounding values.

Therefore, the function does have a local maximum at \(x = 0\). To summarize, if a function climbs to a higher point at a specific spot than nearby points, that spot is termed as a local maximum in mathematical parlance.

Absolute Value Function

The absolute value of a number represents its distance from zero on the number line without considering its direction. In algebra, the absolute value function is denoted by \(|x|\). This function transforms any negative value to a positive and keeps positive values intact.

For instance, the absolute value of -3 is 3, written as \(|-3| = 3\). Thus, absolute value functions create V-shaped graphs when plotted, with a sharp point located at the origin. In our problem, this shape directly impacts how solutions form and behave around specific points.

• The expression \(f(x) = |x|\) for \(0 < |x| \leq 2\) implies the function reflects over the y-axis, retaining positive values.
• Such functions are continuous and have non-negative outputs, essential when evaluating limits and behaviors, especially around crucial points like \(x = 0\).

Function Analysis

Function analysis involves examining how different parts of a function behave, especially across distinct intervals. With piecewise functions, each segment represents a unique formula defining the function behavior over particular ranges.

For this function analysis, you need to observe intervals individually. The behavior of the function at \(0 < |x| \leq 2\) utilizes \(|x|\), indicating linear but symmetric growth on either side of \(x\).

• First, understand that for \(0 < |x| \leq 2\), the function follows a symmetric path as \(x\) approaches zero from positive or negative directions.
• Secondly, analyze the behavior at the pivotal \(x = 0\), where the function deviates to a constant value of 1, altering its behavior and signifying a local extremum.

Through function analysis, gain insight into how each part contributes to the overall structure, helping identify specific points of interest like local maxima or minima.