Problem 117 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f:(-1,1) \rightarrow R\) be a differentiable function such that \(f(0)=-1\) and \(f^{\prime}(0)=1\). Let \(g(x)=[f(2 f(x)+2)]^{2}\). Then \(g^{\prime}(0)=\) [2010] (A) \(-4\) (B) 0 (C) \(-2\) (D) 4 2

Step-by-Step Solution

Answer

The derivative \( g'(0) = -4 \), option (A).

Step 1: Understand the Function

We need to find the derivative of the function \( g(x) = [f(2f(x) + 2)]^2 \) at \( x = 0 \). This requires using the chain rule for differentiation multiple times due to the composition of functions.

Step 2: Differentiate Using the Chain Rule

Let \( h(x) = 2f(x) + 2 \). Then \( g(x) = [f(h(x))]^2 \). The derivative \( g'(x) \) is given by \( g'(x) = 2f(h(x)) \cdot f'(h(x)) \cdot h'(x) \) using the chain rule.

Step 3: Find Derivatives of Inner Functions

Since \( h(x) = 2f(x) + 2 \), compute \( h'(x) = 2f'(x) \). We also have \( f'(0) = 1 \) from the problem statement. These derivatives will help in evaluating \( g'(0) \).

Step 4: Evaluate Components at x=0

Find \( h(0) = 2f(0) + 2 = 0 \). So, \( f(h(0)) = f(0) = -1 \). Also, note that \( f'(h(x)) \) equals \( 1 \) when \( x = 0 \), because both \( h(x) \) and \( h'(x) \) depend directly on \( f(0) \) and \( f'(0) \) respectively.

Step 5: Substitute and Solve for g'(0)

Substitute these into the expression from Step 2: \( g'(0) = 2f(h(0)) \cdot f'(h(0)) \cdot h'(0) = 2(-1)(1)(2) = -4 \). Therefore, \( g'(0) = -4 \).

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used for finding the derivative of a composite function. It illuminates how functions nested within each other can be differentiated. When dealing with composite functions such as \( g(x) = [f(2f(x) + 2)]^2 \), the chain rule helps break down the problem into more manageable parts.

To apply the chain rule, you identify the outer and inner functions. For our given example:

• Outer function: \( u^2 \) where \( u = f(2f(x) + 2) \)
• Inner function: \( f(2f(x) + 2) \)

The derivative of the outer function \( (u^2)' \) is \( 2u \), and the derivative of the inner function would involve further use of the chain rule, defining another nested function.

Using the chain rule iteratively simplifies calculus problems where functions are layered over each other. It's especially handy for handling more complex function compositions, leading to accurate results in differentiating such functions.

Derivative Calculation

Derivative calculation is central to understanding change and rates within a function. To find \( g'(x) \) for the function \( g(x) = [f(2f(x) + 2)]^2 \), the goal is to calculate the rate of change of \( g(x) \) at a specific point, in this case \( x=0 \).

First, you identify the derivative of each part of the compound function. This involves:

• Computing the derivative of the outer function: If \( g(x) = v^2 \), then \( g'(x) = 2v \cdot v' \).
• Calculating the inner derivative \( v = f(2f(x)+2) \), requiring further differentiation by applying chain rule again within the inner part.
• Assembling the derivatives to establish the complete derivative \( g'(x) \).

The calculations involve using known derivative values provided in the exercise, such as \( f'(0) = 1 \), to substitute and find the derivative at \( x=0 \). The result is \( g'(0) = -4 \), a thorough assessment of the shifts in the function \( g(x) \) at the boundary condition given.

Function Composition

Function composition involves linking functions together where the output of one function becomes the input of another. This is a key idea in understanding how complex functions are built and manipulated.

In our example, \( g(x) = [f(2f(x) + 2)]^2 \), the function \( g \) is constructed using the composition of multiple functions:

• \( h(x) = 2f(x) + 2 \)
• \( f(h(x)) = f(2f(x)+2) \)

By understanding the relationships and interactions between these composed functions, one can effectively navigate through each layer to derive the needed calculus properties.

Merging functions in this manner can model sophisticated systems and phenomena mathematically, allowing for exploration and analysis using derivative and chain rule as described in the earlier sections.