Ganitagya Gyanam

  Questions on Maxima and Minima Topic: Maxima and Minima – Calculus | Engineering Mathematics Find the maximum and minimum value of the function f(x, y) = x² + y² − 2x − 4y + 6 . Find the maximum and minimum values of f(x, y) = x² + y² subject to the constraint x + y = 1 . Find the stationary points of f(x, y) = x³ + y³ − 3xy and determine their nature. Find the local maxima, minima, and saddle points of the function f(x, y) = x² − y² . Find the maximum value of u = xy when 2x + 3y = 6 . Find the absolute maximum and minimum values of f(x, y) = x² + y² on the rectangle 0 ≤ x ≤ 2 , 0 ≤ y ≤ 3 . Find the point on the plane x + y + z = 6 closest to the origin. Find the maximum value of f(x, y) = x²y subject to the constraint x² + y² = 1 . Find the minimum value of f(x, y) = x² + y² − 4x − 6y + 13 . Find the stationary point of f(x, y) = x⁴ + y⁴ − 4xy + 1 . Note: These questions cover unconstrained optimization...

Partial derivatives serve as a powerful tool for uncovering the peaks of functions, especially those with multiple variables. when dealing with functions of two variables let us say F(x, y), finding their maximum and minimum points becomes a practical pursuit, led by real-world implications. imagine you're navigating a landscape of Shivalik hills and valleys, or each point representing a potential profit and loss in a business endeavor. to pinpoint the highest peaks of profit and the lowest troughs of loss, partial derivatives help us do just that. First, we hunt for what's known as stationary points- spots where the slope or derivative is zero. By analyzing the sign changes in the partial derivatives around these stationary points, we discern whether they signify maxima, minima, or neither.

Lets understand what is Jacobian. Imagine you have a function that takes several variables as input and gives you a vector (a list of numbers) as output. For example, it could take in the coordinates of a point in space and give you the velocities of particles at that point. Now, let's say you want to understand how small changes in the input variables affect the output vector. The Jacobian matrix helps with exactly that. It's a matrix where each entry represents how much each component of the output vector changes with respect to each input variable. The determinant of this matrix, called the Jacobian determinant or simply the Jacobian, tells you how the volume changes when you transform from the input space to the output space. In simpler terms, it gives you a measure of how much the function "stretches" or "shrinks" space around a particular point. So, in essence, the Jacobian is a way to quantify how a function distorts space when you apply it to a set o...

A Mathematics Exhibition is being conducted in your School and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz tor the audience. Observe the following factor tree and answer the following: 0% Question 1: What will be the value of x? A) 15005 B) 13915 C) 56920 D) 17429 Explanation: 13915. Question 2: What will be the value of y? A) 23 B) 22 C) 11 D) 19 Explanation: 11. Question 3: What will be the value of z? A) 22 B) 23 C) 17 D) 19 Explanation: 23. Question 4: According to Fundamental Theorem ot Arithmetic 13915 is a A) Composite number B) Prime number C) Neither prime nor composite D) Even number Explanation: Composite number Question 5: The prime factorization of 13915 is A) `5 \times 11^3 \times 13^2 ` B) `5\...

 

A seminar is being conducted by an Educational Organisation, where the participants will be educators of different subjects. The number ot participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. 1. In each room the same number of participants are to be seated and all of them being in the same subject, hence maximum number participants that can accommodated in each room are A) 14 B) 12 C) 16 D) 18 ... Answer is B) 12. Show Answer 2. What is the minimum number of rooms required during the event? A) 11 B) 31 C) 41 D) 21 ... Answer is D) 21 Show Answer 3. The LCM of 60, 84 and 108 is A) 3780 B) 3680 C) 4780 D) 4680 ... Answer is A) 3780. Show Answer 4. The product of HCF and LCM of 60, 84 and 108 is A) 55360 B) 35360 C) 45500 D) 45360 ... Answer is D) 45360 Show Answer 5. 108 can be expressed as a product of its primes ...

To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B. 1. What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of Section A or Section B? A) 144 B) 128 C) 288 D) 272 ... Answer is C) 288 Show Answer 2. If the product of two postive integers is equal to the product of their HCF and LCM is true then, the HCF (32, 36) is A) 2 B) 4 C) 6 D) 8 ... Answer is b) 4 Show Answer 3. 36 can be expressed as a product of its primes as A) `2^2  \times 3^2 ` B) `2^1 \times 3^3` C) `2^3 \times 3^1` D) `2^0 \times 3^0` ... Answer is A) 2^2 x 3^2. Show Answer 4. 7x 11x 13 x 15 +15 is a A) Prime number B) compos...