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

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 \(g^{\prime}(0)=\) (A) \(-4\) (B) 0 (C) \(-2\) (D) 4

Step-by-Step Solution

Answer

The value of \( g'(0) \) is 4 (Option D).

Step 1: Understand the Function Composition

We have the function \( g(x) = [f(2f(x) + 2)]^2 \). We need to find the derivative of this function at \( x = 0 \). This requires understanding how the function structure works and requires the chain rule for differentiation.

Step 2: Differentiation Using the Chain Rule

The chain rule states that the derivative of a composition of two functions \( h(x) = u(v(x)) \) is \( h'(x) = u'(v(x))v'(x) \). Here, let \( u(y) = [f(y)]^2 \) and \( v(x) = 2f(x) + 2 \). We then have: \[ g'(x) = 2f(2f(x)+2) imes f'(2f(x)+2) imes (2f'(x)) \]. We will evaluate this at \( x = 0 \).

Step 3: Evaluate Functions at x = 0

To find \( g'(0) \), evaluate each function at \( x = 0 \):- First compute \( f(0) = -1 \).- Then \( v(0) = 2f(0) + 2 = 2(-1) + 2 = 0 \).- So now we evaluate \( f'(0) = 1 \) (given).

Step 4: Use Evaluated Values to Compute Derivative

Using the values calculated:\[ g'(0) = 2 imes f(0)^2 imes f'(0) imes 2f'(0) \= 2 imes (-1)^2 imes 1 imes 2 imes 1 \= 2 imes 1 imes 2 \= 4 \].

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used for differentiating compositions of functions. If you have two functions, say

• an outer function \( u(y) \) and
• an inner function \( v(x) \)

,the chain rule provides a way to differentiate the composite function \[ h(x) = u(v(x)) \].You can think of it like peeling layers of an onion, where you need to understand both functions separately to get the full picture of their combination.With the chain rule, we express the derivative of the composition as:

• \( h'(x) = u'(v(x)) \times v'(x) \)

,allowing us to compute how quickly the output of the composite function changes with respect to changes in \( x \).In our example function \( g(x) = [f(2f(x) + 2)]^2 \), we applied the chain rule by setting

• \( u(y) = [f(y)]^2 \)
• \( v(x) = 2f(x) + 2 \)

,then differentiated with respect to \( x \).This approach allows us to tackle complex functions by breaking them down into simpler parts that are manageable.

Differentiable Function

A function is said to be differentiable if it has a derivative at each point in its domain.This means it can be smoothly graphed without any sudden jumps, breaks, or cusps.Differentiability ensures that the function behaves in a predictable fashion that allows us to calculate the slope at any given point.When you're dealing with a differentiable function like \( f(x) \), it implies that:

• The function has a well-defined tangent at each point.

The slope of this tangent line is what we refer to as the derivative \( f'(x) \).For the function \( f:(-1,1) \rightarrow \mathbb{R} \) in our exercise, knowing that it is differentiable and that \( f'(0) = 1 \) informs us about the behavior of \( f \) at the specific point \( x = 0 \).This information is crucial while applying the chain rule to find \( g'(0) \).

Function Composition

Function composition involves creating a new function by applying one function to the results of another.It is denoted by \( (f \circ g)(x) = f(g(x)) \).Here, the output of the inner function \( g(x) \) becomes the input for the outer function \( f \).In our exercise, the function \( g(x) = [f(2f(x) + 2)]^2 \) is a composition of multiple functions:

• First, the innermost function is \( f(x) \).
• Next, the result from \( f(x) \) is used in the linear function \( 2f(x) + 2 \).
• Finally, \( f(y) \) is squared to get our full composition.

This nested structure is typical in problems requiring differentiation techniques like the chain rule.Function composition allows us to model more complicated processes by combining simpler functions, making them easier to manage mathematically.