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Chapter 7: Problem 29

Statement-1: \(f(x)=x+\cos x\) is strictly increasing. Statement-2: If \(f(x)\) is strictly increasing, then \(f^{\prime}(x)\) may vanish at some finite number of points. (1) Statement \(-1\) is True, Statement \(-2\) is True ; Statement -2 is a correct explanation for Statement \(-1\) (2) Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement \(-1\) is True, Statement \(-2\) is False (4) Statement \(-1\) is False, Statement \(-2\) is True

Short Answer

Expert verified
Option (2) is correct: Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

Step by step solution

01

Differentiate the function

To determine if the function \(f(x) = x + \cos x\) is strictly increasing, first compute its derivative: \[f'(x) = \frac{d}{dx}(x + \cos x) = 1 - \sin x.\]
02

Analyze the derivative

A function is strictly increasing if its derivative is always positive. We need to analyze the expression \(1 - \sin x\). The range of \(\sin x\) is \([-1, 1]\). Thus, the minimum value of \(1 - \sin x\) is \(1 - 1 = 0\) and the maximum value is \(1 - (-1) = 2\). Since \(1 - \sin x\) is never negative, it is always non-negative.
03

Conclusion for Statement 1

Since the derivative \(f'(x)\) is always non-negative and equals zero only at isolated points, \(f(x) \;=\; x + \cos x\) is strictly increasing. Therefore, Statement 1 is true.
04

Evaluate Statement 2

Statement 2 claims that if \(f(x)\) is strictly increasing, then \(f'(x)\) may vanish at some finite number of points. This is correct, as seen in the case of \(f(x) = x + \cos x\), where \(f'(x) = 0\) at isolated points (e.g., \(x = \frac{\pi}{2} + 2k\pi, \;k\in \mathbb{Z}\)). Therefore, Statement 2 is also true.
05

Determine the relationship

Although both statements are true, Statement 2 does not explain Statement 1. Statement 1 is true because \(1 - \sin x\) is always non-negative, not because of the behavior of \(f'(x)\) vanishing at isolated points. Thus, Statement 2 is not a correct explanation for Statement 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

strictly increasing functions
A function is called strictly increasing if, for every pair of inputs where one is larger than the other, the output of the function for the larger input is also larger than for the smaller input.
This concept is crucial in many areas of calculus and mathematics in general.
  • To determine if a function is strictly increasing, we often use its derivative.
    The function is strictly increasing if this derivative is positive for all inputs in the domain.
  • For example, consider the function given in the original exercise: \(f(x) = x + \text{cos} \ x\). To check its monotonicity, we compute its derivative: \(f'(x) = 1 - \text{sin} \ x\).
Since the sine function oscillates between \(-1\) and \(1\), \(1 - \text{sin} \ x\) takes values between 0 and 2.
Because this is never negative, it implies that our function is non-decreasing. However, since it can be zero at isolated points,
the strict inequality isn't strict everywhere, but the function is still commonly classified as strictly increasing if the points of zero derivative are isolated.
function differentiation
Differentiation is a fundamental tool in calculus that helps us analyze the rate at which a function changes.
The derivative of a function at a given point is the slope of the tangent line to the function's graph at that point.
  • To differentiate a sum of functions, we differentiate each term separately.
  • In the exercise, the function is \(f(x) = x + \text{cos} \ x\), and its derivative is found by differentiating each term: \(f'(x) = 1 - \text{sin} \ x\).
    The derivative indicates how the function behaves locally. If the derivative is positive at all points, the function is rising at every point.
Differentiation also helps in identifying critical points where the function’s rate of change is zero or undefined.
Critical points can offer insights into the local minima, maxima, or inflection points of the function.
derivative analysis
Analyzing the derivative of a function is essential to understand its behavior.
Derivative analysis involves looking at the sign and value of a function's derivative to determine properties like monotonicity.
  • If the derivative is positive, the function is increasing in that interval.
    If it is negative, the function is decreasing.
  • In the given exercise, \(1 - \text{sin} \ x\) was analyzed to infer the behavior of \(f(x) = x + \text{cos} \ x\).
  • Sine ranges between \(-1\) and \(1\), hence \(1 - \text{sin} \ x\) ranges from 0 to 2, confirming the derivative is never negative.
    This analysis helps conclude that the function is strictly increasing because the derivative is non-negative and zero only at isolated points.
Evaluating the derivative at various points can also locate where the function flattens out temporarily.
It is also key in verifying other properties such as concavity and convexity.

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