Problem 217 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(p(x)\) be a function defined on \(R\) such that \(p^{\prime}(x)=\) \(p^{\prime}(1-x)\), for all \(x \in[0,1], p(0)=1\) and \(p(1)=41\). Then \(\int_{0}^{1} p(x) d x\) equals \([2010]\) (A) 21 (B) 41 (C) 42 (D) 20

Step-by-Step Solution

Answer

The integral \(\int_{0}^{1} p(x) dx = 21\). Answer: (A) 21.

Step 1: Understand the Function Symmetry

Notice from the condition \(p^{\prime}(x) = p^{\prime}(1-x)\), this implies that the derivative of the function \(p(x)\) is symmetric about \(x = \frac{1}{2}\). This suggests that the function \(p(x)\) itself might also be symmetric around \(x = \frac{1}{2}\).

Step 2: Use Boundary Conditions

We are given that \(p(0) = 1\) and \(p(1) = 41\). This suggests that \(p(x)\) is increasing as \(x\) goes from 0 to 1. Given the symmetry in the derivative, the increase in \(p(x)\) from 0 to 1 mirrors the increase we would expect from \(x = 1\) to \(x = 0\).

Step 3: Evaluate the Integral

Given that \(p(x)\) is symmetric about \(x = \frac{1}{2}\), the average value of \(p(x)\) over the interval \([0, 1]\) can be found by calculating the average of \(p(0)\) and \(p(1)\). Thus, \(\int_0^1 p(x) dx = \frac{p(0) + p(1)}{2} \cdot 1 \). Substitute the given values: \(\int_0^1 p(x) dx = \frac{1 + 41}{2} = 21\).

Key Concepts

Understanding Function Symmetry

In the given exercise, the symmetry of the derivative, expressed by the condition \(p^{\prime}(x) = p^{\prime}(1-x)\), indicates a special property of the function \(p(x)\). This means that the rate of change of the function is symmetric around the midpoint of the interval \([0, 1]\), which is \(x = \frac{1}{2}\). When a function has a symmetric derivative, it often indicates that the function itself might exhibit some symmetric behavior.

• This means that \(p(x)\) might have a balanced or reflective property around \(x = \frac{1}{2}\).
• The implication of this symmetry helps us predict and calculate other function values more easily.
• In this specific problem, recognizing the symmetry in \(p^{\prime}(x)\) helps simplify the process of integration.

To check if \(p(x)\) itself has symmetrical properties, analyze any given or derived conditions, such as boundary conditions, that could validate this symmetry further.

Importance of Boundary Conditions

Boundary conditions are crucial for understanding the behavior of \(p(x)\) over the interval \([0, 1]\). We know that \(p(0) = 1\) and \(p(1) = 41\). These conditions provide fixed values at the endpoints of the interval, which not only describe the boundaries of the function but also hint at its nature.

• The increase from 1 at \(x = 0\) to 41 at \(x = 1\) suggests that \(p(x)\) is increasing over this interval.
• Understanding these boundaries helps in imagining the potential "shape" or "graph" of \(p(x)\).
• Boundary conditions, combined with the symmetry of the derivative, confirm the steady increase of \(p(x)\) matching our understanding of symmetrical properties.

This increase aligns with the symmetrical nature around \(x = \frac{1}{2}\), where any deviation from symmetry would require \(p^{\prime}(x) eq p^{\prime}(1-x)\). Knowing the boundary values allows you to solve for the integral more confidently.

Finding the Average Value

The average value of a function over a given interval is interpreted as a single value that represents the entire area under the curve within that interval. In terms of the integral, it is given by\[ \text{average value} = \frac{1}{b-a} \int_a^b p(x)dx \]In this problem, because the function \(p(x)\) shows symmetry, we smooth out complications by averaging the given boundary values.

• Calculate the average value of \(p(0)\) and \(p(1)\) which are the boundary values.
• The calculation is \(\frac{p(0) + p(1)}{2} = \frac{1 + 41}{2}\).
• This results directly into 21, pointing us towards the integral value over \([0,1]\).

Thus, understanding the average value provides a direct route to calculating the integral, as the midpoint of these symmetrical endpoints encapsulates the entire behavior of \(p(x)\) within this specific interval. This approach not only simplifies the calculation but also illustrates a clear and intuitive notion of symmetry and balance in the function.