Problem 17 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(f(x)=\sin x-\cos x\), the function decreasing in \(0 \leq x \leq 2 \pi\) is \mathrm{\\{} I U P S E A T - 2 0 0 1 ] ~ (a) \(\left[\frac{5 \pi}{6}, \frac{3 \pi}{4}\right]\) (b) \(\left[\frac{\pi}{4}, \frac{\pi}{2}\right]\) (c) \(\left[\frac{3 \pi}{2}, \frac{5 \pi}{2}\right]\) (d) None of these

Step-by-Step Solution

Answer

The function is decreasing in \([\frac{\pi}{4}, \frac{\pi}{2}]\), option (b).

Step 1: Find the derivative of the function

To determine when the function is decreasing, we first need to find the derivative of the function. Given \( f(x) = \sin x - \cos x \), compute the derivative: \( f'(x) = \cos x + \sin x \).

Step 2: Establish the condition for decreasing function

For the function to be decreasing, the derivative must be less than zero, \( f'(x) < 0 \), therefore: \( \cos x + \sin x < 0 \).

Step 3: Solve the inequality

Rewrite the inequality \( \cos x + \sin x < 0 \) using the substitution \( \cos x = \frac{1}{\sqrt{2}}( \cos x + \sin x ) \). Solving \( \cos(x - \frac{\pi}{4}) < 0 \) gives \( \frac{\pi}{4} < x < \frac{5\pi}{4} \).

Step 4: Determine where the function is decreasing in the given interval

Since \( \frac{\pi}{4} < x < \frac{5\pi}{4} \), the range of decreasing intervals in \( 0 \leq x \leq 2\pi \) includes \( [\frac{\pi}{4}, \frac{\pi}{2}] \).

Step 5: Select the correct interval from options

From the intervals given in the options, the interval \( [\frac{\pi}{4}, \frac{\pi}{2}] \) fits the decreasing interval of the function.

Key Concepts

Decreasing Function

A function is said to be decreasing on an interval if as the input value (usually denoted by \(x\)) increases, the output value (\(f(x)\)) decreases. Intuitively, as you move to the right along the \(x\)-axis, the graph of a decreasing function falls. This means that any two points, \(x_1\) and \(x_2\) where \(x_1 < x_2\), the value of the function at \(x_1\) is greater than at \(x_2\):

• \(f(x_1) > f(x_2)\)

For a continuous function, we can use derivatives to determine intervals of increase or decrease. A key idea to remember is:

• A function \(f(x)\) is decreasing when its derivative \(f'(x) < 0\) over an interval.

Using this knowledge, we can find when the function is decreasing within a given domain.

Derivative

Derivatives are a fundamental concept in calculus that provide a way to measure how a function changes. The derivative of a function at a point tells us the slope of the tangent line to the function at that point. In practical terms, the derivative describes how quickly the function's output value is changing at a specific input value.

• Symbolically, if \(f(x)\) is a function, then its derivative is denoted as \(f'(x)\).
• The process of finding the derivative is called differentiation.

With trigonometric functions like \(f(x) = \sin x - \cos x\), the derivative is calculated by differentiating each trigonometric component:

• \(\frac{d}{dx}(\sin x) = \cos x\)
• \(\frac{d}{dx}(-\cos x) = \sin x\)

So, the derivative \(f'(x) = \cos x + \sin x\). This derivative helps us understand the behavior of the function, such as identifying intervals where it decreases.

Trigonometric Functions

Trigonometric functions are periodic functions mainly used to model wave-like and rotational motions. The most common trigonometric functions are \(\sin x\), \(\cos x\), and \(\tan x\). These functions are periodic and oscillate between certain values as \(x\) changes.

• The sine function, \(\sin x\), varies between -1 and 1, completing each cycle as \(x\) progresses through the interval \([0, 2\pi]\).
• Similarly, the cosine function, \(\cos x\), also varies between -1 and 1, completing a full cycle over the same interval.

When dealing with expressions like \(\cos x \) and \( \sin x \), transformations can help simplify calculations:

• For instance, the identity \( \cos(x - \frac{\pi}{4}) = \frac{1}{\sqrt{2}}(\cos x + \sin x) \) is useful in solving inequalities like \( \cos x + \sin x < 0 \).

Understanding these properties is crucial for solving more complex equations involving trigonometric functions.

Inequalities

Inequalities express a relationship where two expressions are not necessarily equal but are related in terms of size.

• An inequality like \(a < b\) indicates that \(a\) is less than \(b\).
• In calculus, inequalities often involve functions and their derivatives to determine intervals of increase or decrease.

To find where the function \(f(x) = \sin x - \cos x\) is decreasing, we solve the inequality \(f'(x) = \cos x + \sin x < 0\). By rewriting the expression:

• Use trigonometric identities to simplify: \(\cos(x - \frac{\pi}{4}) < 0\).
• Solve for \(x\) within the relevant domain, \([0, 2\pi]\).

This process allows us to identify specific intervals where the function's derivative is negative, signifying a decrease in the function's value. Such understanding of inequalities is pivotal in analyzing the behavior of functions within a defined range.