Problem 160 - A Complete Resource Book in Mathematics for JEE Main

Question

The greatest value of \(f(x)=(x+1)^{1 / 3}-(x-1)^{1 / 3}\) on \([0,1]\) is: [2002] (A) 1 (B) 2 (C) 3 (D) \(1 / 3\)

Step-by-Step Solution

Answer

The greatest value is 2 (option B).

Step 1: Understand the problem

We are given a function \(f(x) = (x+1)^{1/3} - (x-1)^{1/3}\) and need to find its greatest value on the interval \([0, 1]\). To do this, we should evaluate the function at points within this interval.

Step 2: Calculate the function value at the endpoints

Evaluate \(f(x)\) at the endpoints of the interval: For \(x=0\):\[ f(0) = (0+1)^{1/3} - (0-1)^{1/3} = 1 + 1 = 2 \] For \(x=1\):\[ f(1) = (1+1)^{1/3} - (1-1)^{1/3} = 2^{1/3} - 0^{1/3} = 2^{1/3} \]

Step 3: Compare the values

From the calculations, for \(x=0\), \(f(0)=2\) and for \(x=1\), \(f(1) = 2^{1/3}\).Since \(2^{1/3} \approx 1.26\), the greatest value occurs at \(x=0\) with \(f(x)=2\).

Step 4: Conclusion

The greatest value of the function \(f(x)=(x+1)^{1 / 3}-(x-1)^{1 / 3}\) on the interval \([0,1]\) is 2, which occurs at \(x=0\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function given a particular input. In this exercise, we are tasked with evaluating the function \(f(x) = (x+1)^{1/3} - (x-1)^{1/3}\) within a specific interval, \([0, 1]\). Before diving into calculations, it's essential to understand what happens to the function when various values of \(x\) are plugged in.

When you substitute a value into the function, follow these steps:

• Replace every occurrence of the variable \(x\) with the given number.
• Perform the arithmetic operations in the correct order, generally following the order of operations: parentheses, exponents, multiplication, division, addition, and subtraction.

In this problem, first substitute \(x = 0\) into the function to get \(f(0) = 2\). Next, substitute \(x = 1\) to get \(f(1) = 2^{1/3} \approx 1.26\). These evaluations allow us to compare the outputs to determine which one gives the largest value over the interval. Function evaluation at critical points like these often forms the basis of solving greatest value problems.

Greatest Value Problem

The greatest value problem involves finding the largest output of a function within a specified range or interval. In mathematical terms, this is often referred to as the maximum value of the function on that interval. Solving this problem typically involves evaluating the function at key points: the endpoints of the interval and any critical points within it.

Critical points are those where the derivative of the function is either zero or undefined. These points can signal a possible maximum or minimum value of the function. In this exercise, the function is analyzed primarily at the endpoints \([0, 1]\) because it's simpler due to the lack of other distinct critical points in this interval for the given function type.

To find the greatest value of \( f(x) = (x+1)^{1/3} - (x-1)^{1/3} \), we calculate \( f(x) \) at both endpoint values. We found \( f(0) = 2 \) and \( f(1) \approx 1.26 \). Comparing these values, we determine that the maximum occurs at \(x = 0\) with \(f(x) = 2\). This step confirms that the greatest value in the specified interval is 2, aligning with the solution's outcome.

Interval Analysis

Interval analysis is a technique used to study the behavior of functions over specific intervals. It helps identify potential maximum and minimum values of a function by zeroing in on critical regions. With functions like \((x+1)^{1/3} - (x-1)^{1/3}\), we examine the function within predefined boundaries, here being \([0, 1]\).

By analyzing the function's behavior at the interval's endpoints and any critical points within it, we can identify the extremes of the function's output. Here, the endpoints \(x = 0\) and \(x = 1\) are examined. This method directly leads to concluding that the function's greatest value on the interval falls at one of these points.

When conducting interval analysis, it's helpful to:

• Evaluate the function at all potential maximum or minimum points, including endpoints.
• Compare values to determine which point yields the greatest result.

Through this systematic approach, interval analysis confirms that the greatest value of \(f(x)\) in this scenario is indeed at \(x = 0\) with a result of 2.