Mathematics Exam > Mathematics Questions > What is the saddle point?a)Point where functi...
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What is the saddle point?
- a)Point where function has maximum value
- b)Point where function has minimum value
- c)Point where function has zero value
- d)Point where function neither have maximum value nor minimum value
Correct answer is option 'D'. Can you explain this answer?
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What is the saddle point?a)Point where function has maximum valueb)Poi...
Saddle point is a point where function have neither maximum nor minimum value.
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What is the saddle point?a)Point where function has maximum valueb)Poi...
The saddle point refers to a point on a function's graph where the function neither has a maximum value nor a minimum value. It is a critical point where the function changes concavity, similar to how a saddle changes its shape. In other words, a saddle point is a point of inflection on a function's graph.
Definition:
A saddle point occurs at a point (x, y) on a function's graph where the function has a horizontal tangent line (slope is zero) but changes concavity. Mathematically, it is a point where the function's second derivative changes sign.
Graphical Representation:
To visualize a saddle point on a graph, imagine a curve that initially goes upwards, then flattens out at the saddle point, and finally goes downwards. At the saddle point, the curve changes from being concave up to concave down or vice versa.
Example:
Let's consider the function f(x, y) = x^2 - y^2. To find the saddle point, we need to find the critical points where the partial derivatives are zero. Taking the partial derivatives, we get:
∂f/∂x = 2x = 0
∂f/∂y = -2y = 0
Solving these equations, we find that the critical point is (0, 0). To determine if it is a saddle point, we need to examine the second-order partial derivatives. Taking the second partial derivatives, we get:
∂²f/∂x² = 2
∂²f/∂y² = -2
At the critical point (0, 0), the second partial derivatives have opposite signs, indicating a change in concavity. Therefore, (0, 0) is a saddle point.
Conclusion:
In summary, a saddle point is a point on a function's graph where the function has a horizontal tangent line but changes concavity. It is neither a maximum nor a minimum point. Saddle points are important in optimization problems and can have significant implications in various fields such as economics, physics, and engineering.
Definition:
A saddle point occurs at a point (x, y) on a function's graph where the function has a horizontal tangent line (slope is zero) but changes concavity. Mathematically, it is a point where the function's second derivative changes sign.
Graphical Representation:
To visualize a saddle point on a graph, imagine a curve that initially goes upwards, then flattens out at the saddle point, and finally goes downwards. At the saddle point, the curve changes from being concave up to concave down or vice versa.
Example:
Let's consider the function f(x, y) = x^2 - y^2. To find the saddle point, we need to find the critical points where the partial derivatives are zero. Taking the partial derivatives, we get:
∂f/∂x = 2x = 0
∂f/∂y = -2y = 0
Solving these equations, we find that the critical point is (0, 0). To determine if it is a saddle point, we need to examine the second-order partial derivatives. Taking the second partial derivatives, we get:
∂²f/∂x² = 2
∂²f/∂y² = -2
At the critical point (0, 0), the second partial derivatives have opposite signs, indicating a change in concavity. Therefore, (0, 0) is a saddle point.
Conclusion:
In summary, a saddle point is a point on a function's graph where the function has a horizontal tangent line but changes concavity. It is neither a maximum nor a minimum point. Saddle points are important in optimization problems and can have significant implications in various fields such as economics, physics, and engineering.
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What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer?
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What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer?.
What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the saddle point?a)Point where function has maximum valueb)Point where function has minimum valuec)Point where function has zero valued)Point where function neither have maximum value nor minimum valueCorrect answer is option 'D'. Can you explain this answer?.
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