Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)=\cos x\) and \(g(x)=[x+2]\), where \([.]\) denotes the greatest integer function. Then, \((\text { gof })^{\prime}\left(\frac{\pi}{2}\right)\) is (A) 1 (B) 0 (C) \(-1\) (D) does not exist

Step-by-Step Solution

Answer

Option (B) 0.

Step 1: Define Composite Function

The task is to find the derivative of the composite function \((g \circ f)(x)\) at \(x = \frac{\pi}{2}\). The composite function \(g \circ f\) refers to plugging \(f(x)\) into \(g(x)\), so \((g \circ f)(x) = g(f(x)) = g(\cos x)\).

Step 2: Evaluate Inner Function at \(x = \frac{\pi}{2}\)

First, determine \(f\left(\frac{\pi}{2}\right)\). Since \(f(x) = \cos x\), we have \(f\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0\).

Step 3: Apply Outer Function to Output

Next, calculate \(g(0)\) using the greatest integer function. \(g(x) = [x+2]\), substituting in \(x = 0\), gives \(g(0) = [0 + 2] = [2]\). The greatest integer less than or equal to 2 is 2, so \(g(0) = 2\).

Step 4: Differentiate Composite Function

To find \((g \circ f)'(x)\), calculate the derivative. The greatest integer function is constant between integers, so its derivative is zero except at jump discontinuities. Therefore, the derivative of \(g(f(x))\) with respect to \(x\) is also zero at points where \(f(x)\) is not an integer, which includes \(x = \frac{\pi}{2}\).

Step 5: Conclude Value of Derivative

Since the derivative of \(g(f(x))\) with respect to \(x\) is zero at points where the output is not at a jump discontinuity, we have \((g \circ f)'\left(\frac{\pi}{2}\right) = 0\). Conclude that this corresponds to option (B).

Key Concepts

Composite Function

In calculus, a composite function is formed when one function is applied to the results of another function. Imagine having two functions, say \( f(x) \) and \( g(x) \). When you create a composite function, you are essentially feeding the output of \( f(x) \) directly into \( g(x) \). You will see this notation written as \((g \circ f)(x)\), which means \( g(f(x)) \).
For example, if you have \( f(x) = \cos x \) and \( g(x) = [x+2] \), as in our exercise, the composite function \((g \circ f)(x)\) would be \( g(\cos x) \).
This concept is crucial because it allows us to calculate more complex relationships and behaviors by combining simpler functions.

• Composite functions are formed by taking the output of one function and using it as the input for another.
• Notation: \((g \circ f)(x)\) = \( g(f(x)) \).
• Useful in modeling and solving real-world problems that involve combining processes or operations.

Greatest Integer Function

The greatest integer function, also known as the floor function, is a mathematical function denoted by \([x]\). This function takes a real number \(x\) and rounds it down to the nearest integer, i.e., it gives you the largest integer less than or equal to \(x\).
For example, \([3.7] = 3\) and \([-1.5] = -2\). The greatest integer function can be quite useful when you need to discretize a continuous value, such as rounding down prices or scores.

• Symbol: \([x]\).
• Rounds any real number down to the nearest integer.
• Important in fields requiring utmost precision and integer results.

When incorporating it into a composite function, like \(g(x) = [x+2]\), it affects the function's continuity.
In this particular function, \(g(x)\) remains constant between integers because the floor function "jumps" to a new value only at the integers.

Derivative

The derivative is a fundamental concept in calculus, representing the rate of change or the slope of a function at any given point. Essentially, the derivative of a function \( f(x) \) tells us how \( f(x) \) changes as \( x \) changes. This is written as \( f'(x) \).
In cases with functions like the greatest integer function, the derivative can exhibit unique characteristics. The greatest integer function is constant between integers and has jump discontinuities at integer points. This implies that its derivative is zero wherever there is no discontinuity.
From the original exercise, since the derivative of the composite function \((g \circ f)'\left(\frac{\pi}{2}\right)\) isn't at a point of discontinuity, it concludes as zero.

• Represents the rate of change of a function.
• Notation: \( f'(x) \).
• Important in understanding and modeling dynamic systems and change.

Understanding how derivatives work in different types of functions can help in predicting and analyzing behaviors in various practical applications, like physics, engineering, and economics.