Minimum d occurs at minimum d^2

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In summary: Therefore, if ##d_1 < d_2##, then ##d_1^2 < d_2^2##. This means that if we can find a point where ##d_1 < d_2##, then we know that the minimum value of ##d_1^2## must also be less than the minimum value of ##d_2^2##. Therefore, the point that minimizes ##d_1## will also minimize ##d_1^2##, meaning it is the same point as the point that minimizes ##d_2##.
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brycenrg
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Homework Statement


Find the point on the parabola y^2 = 2x that is closest to the point (1,4)

Homework Equations


d = ((x-1)^2 + (y-4)^2)^1/2
x=y^2/2

The Attempt at a Solution


My attempt involves subbing in x in the d equation and differentiating it. I can't get the same answer as the book because they differentiate d^2 = not d = to make it easier to solve.
They state "you should convince yourself the minimum of d occurs at the same point as the minimum of d^2, but d^2 is easier to work with"

How do i know it occurs at the same point?
 
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Try to find positive ##d_1 < d_2## with ##d_1^2 > d_2^2##
 
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brycenrg said:

Homework Statement


Find the point on the parabola y^2 = 2x that is closest to the point (1,4)

Homework Equations


d = ((x-1)^2 + (y-4)^2)^1/2
x=y^2/2

The Attempt at a Solution


My attempt involves subbing in x in the d equation and differentiating it. I can't get the same answer as the book because they differentiate d^2 = not d = to make it easier to solve.
They state "you should convince yourself the minimum of d occurs at the same point as the minimum of d^2, but d^2 is easier to work with"

How do i know it occurs at the same point?
If ##0<a<b## then ##a\cdot a < a \cdot b < b\cdot b##.
 
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FAQ: Minimum d occurs at minimum d^2

What does the phrase "minimum d occurs at minimum d^2" mean?

The phrase "minimum d occurs at minimum d^2" refers to a mathematical concept known as the minimum distance problem. It means that the shortest distance between two points, known as d, will also correspond to the smallest possible value for d squared.

Can you provide an example of when minimum d occurs at minimum d^2?

One example of when minimum d occurs at minimum d^2 is when calculating the shortest distance between two points on a graph. The value for d will also be the smallest possible value for d squared.

Why is it important to understand the concept of minimum d occurs at minimum d^2?

Understanding the concept of minimum d occurs at minimum d^2 is important in various fields such as mathematics, physics, and engineering. It helps in solving problems related to distance and optimization, and can also be applied in real-world scenarios such as finding the shortest route for navigation or minimizing costs in manufacturing processes.

Is the phrase "minimum d occurs at minimum d^2" always true?

Yes, the phrase "minimum d occurs at minimum d^2" is always true. This is because the value for d squared will always be smaller than or equal to the value for d. So, if d is at its minimum, then d squared will also be at its minimum.

How is the concept of minimum d occurs at minimum d^2 related to the Pythagorean theorem?

The concept of minimum d occurs at minimum d^2 is closely related to the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This can be applied to finding the shortest distance between two points, where the hypotenuse represents d and the other two sides represent d squared.

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