Problem 146
Question
In the following questions an Assertion (A) is given, followed by a Reason
(R). Mark your responses from the following options:
(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct
explanation for Assertion(A)
(B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct
explanation for Assertion(A)
(C) Assertion(A) is True, Reason(R) is False
(D) Assertion(A) is False, Reason(R) is True
Assertion: The function \(f(x)=\frac{\sin x}{x}\) is decreasing in the interval
\(\left(0, \frac{\pi}{2}\right)\)
Reason: \(\tan x>x\) for \(0
Step-by-Step Solution
Verified Answer
(B) Assertion and Reason are true, but Reason doesn't explain Assertion.
1Step 1: Understand the Given Assertion
The assertion states that the function \( f(x) = \frac{\sin x}{x} \) is decreasing in the interval \( \left(0, \frac{\pi}{2}\right) \). To verify this, we need to examine the derivative of the function and determine its monotonicity in the specified interval.
2Step 2: Differentiate the Function
To examine if the function is decreasing, calculate the derivative: \[ f(x) = \frac{\sin x}{x} \]Using the quotient rule, the derivative is given by: \[ f'(x) = \frac{x \cos x - \sin x}{x^2} \] Evaluate if \( f'(x) \) is negative over the interval \( \left(0, \frac{\pi}{2}\right) \) to determine if the function is decreasing.
3Step 3: Analyze the Derivative
For \( f(x) \) to be decreasing, \( f'(x) \) should be less than 0. Now, \( x \cos x - \sin x < 0 \) simplifies to \( x \cos x < \sin x \). This inequality holds since for small positive \( x \), \( \sin x \approx x \), and \( \cos x < 1 \), thus \( x \cos x < \sin x \). Hence, \( f(x) \) is indeed decreasing in the interval \( \left(0, \frac{\pi}{2}\right) \). The assertion is true.
4Step 4: Examine the Given Reason
The reason states \( \tan x > x \) for \( 0 < x < \frac{\pi}{2} \). Consider the function \( g(x) = \tan x - x \). We know \( g(0) = \tan 0 - 0 = 0 \), and \( g'(x) = \sec^2 x - 1 > 0 \) for \( 0 < x < \frac{\pi}{2} \). Thus, \( g(x) > 0 \). Hence, \( \tan x > x \). The reason is true.
5Step 5: Evaluate Relationship Between Assertion and Reason
Now, determine if the reason (\( \tan x > x \)) is a correct explanation for the assertion (\( \frac{\sin x}{x} \) is decreasing). The reason does not directly explain the behavior of \( \frac{\sin x}{x} \) being decreasing. While both are true individually, the reason does not contribute to explaining why the assertion is true. Hence, the reason is not a correct explanation for the assertion.
Key Concepts
Derivative AnalysisMonotonicityTrigonometric Inequalities
Derivative Analysis
Derivative analysis is a fantastic tool we use in mathematics to understand how a function behaves. It tells us whether a function is increasing, decreasing, or staying constant at any given point or interval.
To perform this analysis, we differentiate the function. In our exercise, we looked at the function \( f(x) = \frac{\sin x}{x} \). By applying the quotient rule, we found its derivative:
\[ f'(x) = \frac{x \cos x - \sin x}{x^2} \]
This derivative helps us determine where the function is increasing or decreasing. By examining the sign of \( f'(x) \), we can see how the slope (or steepness) of the function changes over the interval. A negative derivative implies a decreasing function.
Understanding derivatives not only helps us graph functions more accurately but also gives insights into optimization problems and motion in physics.
To perform this analysis, we differentiate the function. In our exercise, we looked at the function \( f(x) = \frac{\sin x}{x} \). By applying the quotient rule, we found its derivative:
\[ f'(x) = \frac{x \cos x - \sin x}{x^2} \]
This derivative helps us determine where the function is increasing or decreasing. By examining the sign of \( f'(x) \), we can see how the slope (or steepness) of the function changes over the interval. A negative derivative implies a decreasing function.
Understanding derivatives not only helps us graph functions more accurately but also gives insights into optimization problems and motion in physics.
Monotonicity
Monotonicity is all about a function maintaining a consistent direction, either always going up (increasing) or going down (decreasing). In the exercise we have, the focus was on proving whether the function \( f(x) = \frac{\sin x}{x} \) maintains a decreasing behavior in the interval \( \left(0, \frac{\pi}{2}\right) \).
To determine this, we considered the derivative we found: \( f'(x) = \frac{x \cos x - \sin x}{x^2} \). When we simplify, we get that \( x \cos x < \sin x \), confirming the derivative is negative in this interval. Therefore, the function indeed persists in a decreasing manner.
The concept of monotonicity is crucial in calculus as it gives us a clear picture of how functions behave. Knowing whether a function is strictly increasing or decreasing helps in understanding limits, solving equations, and analyzing trends in data.
To determine this, we considered the derivative we found: \( f'(x) = \frac{x \cos x - \sin x}{x^2} \). When we simplify, we get that \( x \cos x < \sin x \), confirming the derivative is negative in this interval. Therefore, the function indeed persists in a decreasing manner.
The concept of monotonicity is crucial in calculus as it gives us a clear picture of how functions behave. Knowing whether a function is strictly increasing or decreasing helps in understanding limits, solving equations, and analyzing trends in data.
Trigonometric Inequalities
Trigonometric inequalities involve mathematical expressions that use trigonometric functions such as sine, cosine, and tangent. In the exercise, we evaluated the inequality \( \tan x > x \) for \( 0 < x < \frac{\pi}{2} \). To prove this, we analyzed a related function \( g(x) = \tan x - x \).
By evaluating this derivative, \( g'(x) = \sec^2 x - 1 \), we identify it as positive in our interval, implying that the function consistently increases. Since \( g(0) = 0 \), and because \( g(x) \) increases, \( g(x) > 0 \) is valid throughout the interval, confirming the implication \( \tan x > x \).
Working with trigonometric inequalities allows us to solve complex problems involving angles and triangles. It's a helpful skill that connects to many aspects of math, physics, and engineering, providing practical solutions in wave physics, oscillations, and other analytical tasks.
By evaluating this derivative, \( g'(x) = \sec^2 x - 1 \), we identify it as positive in our interval, implying that the function consistently increases. Since \( g(0) = 0 \), and because \( g(x) \) increases, \( g(x) > 0 \) is valid throughout the interval, confirming the implication \( \tan x > x \).
Working with trigonometric inequalities allows us to solve complex problems involving angles and triangles. It's a helpful skill that connects to many aspects of math, physics, and engineering, providing practical solutions in wave physics, oscillations, and other analytical tasks.
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