Problem 107 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f^{\prime \prime}(x)>0 \forall x \in R\) and \(g(x)=f(2-x)+f(4+x)\). Then, \(g(x)\) is increasing in (A) \((-\infty,-1)\) (B) \((-\infty, 0)\) (C) \((-1, \infty)\) (D) None of these

Step-by-Step Solution

Answer

Option (C) \((-1, \infty)\).

Step 1: Understanding the Given Functions

The problem provides two conditions: the second derivative \( f^{\prime \prime}(x) > 0 \) for all \( x \) implies that \( f(x) \) is a convex function. The function \( g(x) = f(2-x) + f(4+x) \) is what we are analyzing, which is a combination of transformations on \( f(x) \).

Step 2: Find the Derivative of \( g(x) \)

To understand the behavior of \( g(x) \), we first find its derivative: \[ g^{\prime}(x) = \frac{d}{dx}[f(2-x) + f(4+x)] = f^{\prime}(2-x)(-1) + f^{\prime}(4+x)(1) = -f^{\prime}(2-x) + f^{\prime}(4+x). \]

Step 3: Analyze the Sign of \( g^{ ext{'}}(x) \)

We want to determine where \( g^{\prime}(x) > 0 \), as this will show us where \( g(x) \) is increasing. Simplifying the expression, \[ g^{\prime}(x) = f^{\prime}(4+x) - f^{\prime}(2-x). \] Since \( f^{\prime \prime}(x) > 0 \) for all \( x \), \( f^{\prime}(x) \) is increasing.

Step 4: Conclusion for Increasing Regions

Given the increase of \( f^{\prime}(x) \), for \( x > -1 \), the term \( f^{\prime}(4+x) \) will be greater than \( f^{\prime}(2-x) \). Hence, \( g^{\prime}(x) > 0 \) for \( x > -1 \), making \( g(x) \) an increasing function in this interval.

Key Concepts

Second Derivative Test

The Second Derivative Test is a pivotal concept in calculus that helps determine the concavity of functions and their critical points. When a function's second derivative, denoted as \( f''(x) \), is positive for all \( x \), it indicates that the function is convex. This means the graph of the function curves upwards, similar to a cup shape. A convex function's slope constantly increases as you move along the graph.
In the original exercise, we know \( f''(x) > 0 \) for all \( x \). This confirms that \( f(x) \) is a convex function. This property is critical because it suggests that the function's first derivative, \( f'(x) \), is an increasing function. By applying the second derivative test, we have a clear understanding that the original function \( f(x) \) maintains an upward curving pattern without intervals of decrease. This contributes directly to analyzing when a transformation of this function, like \( g(x) \) in the exercise, is increasing.

Function Transformation

Function transformations involve changing a function's position or shape via operations like shifts, stretches, or reflections. In the context of the original exercise, the function \( g(x) = f(2-x) + f(4+x) \) involves transforming \( f(x) \) in two distinct ways:

• \( f(2-x) \) implies a horizontal shift. The expression \( 2-x \) indicates a reflection of \( f(x) \) about the vertical line \( x=1 \), followed by a shift.
• \( f(4+x) \) means the function is shifted to the left by 4 units along the x-axis.

By examining these transformations, we realize that the behavior of \( g(x) \) depends on combining these shifts. It transforms the way \( f(x) \) is viewed and affects its growth and in this case, through the original exercise's lens, helps us determine when \( g(x) \) increases.

Derivative Analysis

Derivative analysis involves examining the first derivative, \( g'(x) \), to understand the function's rate of change. For \( g(x) \) to be increasing, \( g'(x) \) must be positive. Calculating \( g'(x) \) as -\( f'(2-x) \) + \( f'(4+x) \) involves differentiating the transformed function \( g(x) \).
Given \( f'(x) \) is an increasing function (due to \( f''(x) > 0 \)), \( f'(4+x) \) is always larger than \( f'(2-x) \) when \( x > -1 \). Therefore, \( g'(x) > 0 \) for \( x > -1 \), showing that \( g(x) \) increases in this region. The critical point here is understanding that with an increasing \( f'(x) \), the component \( f'(4+x) \) grows faster than \( f'(2-x) \), resulting in a positive trend for \( g'(x) \). This derivative comparison clarifies where \( g(x) \) increases, telling us exactly about \( x > -1 \).