Concavity, Convexity and Point of Inflexion-Previous year questions

Previous Year Questions on Concavity, Convexity and Point of Inflexion

Concavity, Convexity and Point of Inflexion are fundamental topics in mathematics, bridging geometry and calculus. Historically, their study has provided insights into curve behavior and optimization problems. These topics are crucial in understanding Mathematics and particularly relevant for Differential Calculus.

Vidyasagar University

2023-24 (NEP)

Find the range of values of \(x\) for which \(y=x^{4}-6x^{3}+12x^{2}+5x+7 \) is concave upward. [2]

2023-24 (CBCS)

Show that abscissa of the points of inflexion on the curve \(y^{2}=f(x)\) satisfying \(\big[f(x) \big]^{2}=2f(x)f^{\prime}(x)\). [2]

2022-23 (CBCS)

No Questions

2021-22 (CBCS)

Show that the curve \(re^{\theta}=a(1+\theta) \) has no point of inflexion. [4]
Find the point of inflexion on the curve \(r(\theta^{2}-1)=a\theta^{2} \). [3]

2020-21 (CBCS)

Find the point of inflexion on the function \( f(x)=3x^{4}-8x^{3}\). [4]

2019-20 (CBCS)

Find the point of inflexion on the curve \(\big(\theta^{2}-1\big)r=a\theta^{2}\). [5]

2018-19 (CBCS)

Find the range of values of \(x\) for which \begin{align*} y=x^{4}-6x^{3}+12x^{2}+5x+7 \end{align*} is concave upward. [2]
Show that abscissa of the points of inflexion on the curve \(y^{2}=f(x)\) satisfying \(\big[f(x) \big]^{2}=2f(x)f^{\prime}(x)\). [2]
If \(f(x)=ax^{3}+3bx^{2}\). Find \(a\) and \(b\) so that \((1,-2)\) is a point of inflexion of \(f\). [3]

2017-18 (CBCS)

Define point of inflexion of a curve. [2]
Show that the curve \begin{align*} y=3x^{5}-40x^{3}+3x-20 \end{align*} is concave upwards for \(-2\lt x\lt 0 \) and \(2\lt x\lt \infty \) but convex upwards for \(-\infty\lt x\lt -2 \) and \(0\lt x\lt 2 \). Also show that \(x=-2,0,2\) are its points of inflexion. [2+2+1]

FAQs

What is the significance of Concavity and Convexity?
These concepts determine the curvature of a function and are vital for understanding Differential Calculus.
How is the Point of Inflexion identified?
It is determined where the curve changes its concavity, often studied in Differential Calculus.
Are Concavity and Convexity relevant to optimization?
Yes, they are crucial in optimization problems and appear frequently in Mathematics.
Where can I practice related questions?
Practice is available through Linear Algebra and Classical Algebra.
How are these topics visualized?
Graphical representations are used, and insights can be found in Linear Algebra.
What resources cover these topics extensively?
Refer to our Suggested Books for comprehensive coverage.
What are the real-world applications?
Applications include physics and engineering, which rely heavily on principles from Differential Calculus.
What are the key mathematical tools for studying these topics?
Tools include calculus and algebra, highlighted in Classical Algebra.
Where can I find examples?
Examples are available in our Mathematics Questions section.
Why are these concepts essential in mathematics?
They form the foundation for curve analysis and optimization, as covered in Differential Calculus.