Question:
If f is monotonic in [a, b], then f is
If f is monotonic in [a, b], then f is
Show Hint
Monotonicity and Continuity are the two most common sufficient conditions for integrability.
Updated On: Jul 3, 2026
- not bounded in [a, b]
- not integrable in [a, b]
- integrable in [a, b]
- bounded in (a, b) only
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Concept
This is a fundamental theorem stating that monotonicity is a sufficient condition for integrability.
Step 2: Meaning
A monotonic function (either always increasing or always decreasing) on a closed interval $[a,b]$ is always bounded by its values at the endpoints.
Step 3: Analysis
For any monotonic function, the difference between the Upper and Lower sums can be made arbitrarily small by refining the partition, satisfying the Riemann criterion.
Step 4: Conclusion
Therefore, every monotonic function on $[a,b]$ is integrable. Final Answer: (C)
This is a fundamental theorem stating that monotonicity is a sufficient condition for integrability.
Step 2: Meaning
A monotonic function (either always increasing or always decreasing) on a closed interval $[a,b]$ is always bounded by its values at the endpoints.
Step 3: Analysis
For any monotonic function, the difference between the Upper and Lower sums can be made arbitrarily small by refining the partition, satisfying the Riemann criterion.
Step 4: Conclusion
Therefore, every monotonic function on $[a,b]$ is integrable. Final Answer: (C)
Was this answer helpful?
0
0
Top AP ECET BSc Mathematics Mathematics Questions
- The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}
- If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $
- The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
- Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?
- The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
View More Questions
Top AP ECET BSc Mathematics Number Theory Questions
Top AP ECET BSc Mathematics Questions
- The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}
- If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $
- The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
- Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?
- The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
View More Questions