Problem 4 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

The function \(f(x)=x+\cos x\) is [DCE-2002] (a) Always increasing (b) Always decreasing (c) Increasing for certain range of \(x\) (d) None of these

Step-by-Step Solution

Answer

The function is always increasing. (Option a)

Step 1: Understand the Problem

We are given the function \( f(x) = x + \cos x \) and need to determine in which range of \( x \) the function is increasing, decreasing, or neither.

Step 2: Find the Derivative

To determine whether the function is increasing or decreasing, we first find its derivative which represents the function's instantaneous rate of change. The derivative of \( f(x) = x + \cos x \) is given by: \[ f'(x) = \frac{d}{dx}(x) + \frac{d}{dx}(\cos x) = 1 - \sin x \]

Step 3: Analyze the Derivative

For a function to be increasing, its derivative must be greater than zero, and for it to be decreasing, the derivative must be less than zero.Thus, we look for the values of \( x \) where \( f'(x) > 0 \), which gives:\[ 1 - \sin x > 0 \]\[ \sin x < 1 \]This condition is true for all real \( x \) because \( \sin x \) never exceeds 1.

Step 4: Conclusion

Since the derivative \( 1 - \sin x \) is always equal to zero or greater (it can never be less than zero), the function \( f(x) = x + \cos x \) is always increasing.

Key Concepts

Increasing Function

An increasing function is one where, as you move along the x-axis from left to right, the value of the function either stays the same or rises. In simpler terms, the graphs of increasing functions do not dip. They either move straight or go upward.

• If the derivative of a function is positive ( \(f'(x) > 0\)), the function is strictly increasing.
• If the derivative is zero, the function is constant.

To check if our function, \(f(x) = x + \cos x\), is increasing, we can find its derivative: \[f'(x) = 1 - \sin x\]Since \( \sin x\) is always less than or equal to 1, the derivative \(1 - \sin x\) is zero or positive, ensuring that the function is increasing for all real numbers. This means every point on the function's graph stays the same height or rises.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are fundamental in mathematics. They describe wave patterns and circles and are periodic, meaning they repeat their values in regular intervals.

• Cosine Function: Represents the x-coordinate of a point on a unit circle as the circle is traversed at consistent angle intervals. It oscillates between -1 and 1.
• Sine Function: Similar to cosine, but it represents the y-coordinate. The sine also oscillates between -1 and 1.

In our function \(f(x) = x + \cos x\), the behavior of the cosine component adds periodic oscillations to the linear component \(x\). This oscillation, however, doesn't affect whether the function is increasing over the long term because the problematic part is merely an additive part that won't overtake the linear x-term.

Differentiation Techniques

Differentiation is a method from calculus that helps us find the rate at which a quantity changes. It's like a mathematical microscope, zooming in on the small changes.

• Basic Differentiation: The derivative of \(x\), a simple linear term, is 1.
• Derivative of Trigonometric Functions: Using known rules, the derivative of \(\cos x\) is \(-\sin x\).

To find \(f'(x)\) for our function \(f(x) = x + \cos x\), we apply these rules:- Evaluate the derivative of each part separately: - For \(x\), it's straightforward: \( \frac{d}{dx}(x) = 1 \). - For \(\cos x\), apply the trigonometric rule: \( \frac{d}{dx}(\cos x) = -\sin x \).Combine these results to obtain \(f'(x) = 1 - \sin x\). This calculation shows how the derivatives specify the velocity or rate of change of the function's slope across its domain.