Find all functions in 1. quadrant which tangts form triangles

In summary, the task is to find all functions for which any tangent in the first quadrant forms a triangle with a constant surface P. The approach involves using a differential equation and solving it, but the poster is unsure of how to begin. The tangent is a linear function and is equal to the derivative of the function. The poster has not made much progress on the problem and is seeking help.
  • #1
skrat
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Homework Statement


Find all functions for which any tangent in first quadrant "forms" a triangle with constant surface P. (You can assume that y'<0)



Homework Equations





The Attempt at a Solution


Now, I know I should somehow get to differential equation and then solve it but, I haven't got a clue on how to start?

Tangent is a linear function with ##k=\frac{y_2-y_1}{x_2-x_1}## and also equal to ##k=y^{'}(x)##..

That is in fact everything I have so far. I've been staring and this problem for quite a while now. Please help.
 
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  • #2
Take a point on the curve, y = f(x). Where does the tangent at that point intersect the axes?
 

1. What is the purpose of finding all functions in the 1st quadrant which form tangents for triangles?

The purpose of this task is to identify all possible functions that can be used to form triangles in the 1st quadrant, based on the tangent line that the function creates.

2. How do you determine which functions form tangents for triangles in the 1st quadrant?

This can be determined by finding the slope of the tangent line at any point on the function and ensuring that it falls within the range of possible slopes for a triangle in the 1st quadrant (0 to 1).

3. Can any type of function form a tangent for a triangle in the 1st quadrant?

No, the function must have a positive slope and be continuous in the 1st quadrant in order to form a tangent for a triangle.

4. Are there any limitations to finding all functions in the 1st quadrant which form tangents for triangles?

One limitation is that the function must be defined for all values in the 1st quadrant. Additionally, there may be functions that form tangents for triangles but are not continuous or have a negative slope, which would not be included in the results.

5. How can knowing all functions in the 1st quadrant which form tangents for triangles be useful?

This information can be useful in understanding the geometry and properties of triangles in the 1st quadrant. It can also be helpful in solving geometric problems and equations involving tangents and triangles.

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