Graphical meaning of tangent in optimization problem

In summary, the solution to a trivial optimization problem involves finding the value of x2 that minimizes y(x2)/(x2-x1). This can be graphically represented by the tangent line shown in the figure. However, the logical steps behind this solution are not clear, and more information on y(x) or x1 is needed to fully understand it. Assuming y = x2, it is possible to minimize x2/(x-x1) by using the rule for deriving a fraction and checking if it can be equal to 0.
  • #1
nigels
36
0
In a trivial optimization problem, when seeking the value of x2 that minimizes y(x2)/(x2-x1), the solution is graphically given by the tangent line shown in the figure.

I'm having a lot of difficulty understanding why this is true, i.e., the logical steps behind the equivalence supporting the solution, either via calculus, algebraic, or geometric reasoning.
oft.png
 
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  • #2
Are you sure that you have given all information? The curve looks suspiciously like y = x2.
 
  • #3
@Svein: Sorry, x1 and x2 mean x_1 and x_2.
 
  • #4
nigels said:
seeking the value of x2 that minimizes y(x2)/(x2-x1)
If we have no information on y(x) or on x1, it is impossible to answer. If we assume y = x2, it is at least possible: Minimize x2/(x-x1). Assuming that it exists, use the rule for deriving a fraction and see if it can be equal to 0.
 
  • #5
I think you need this
 

1. What is the graphical meaning of tangent in optimization problem?

The graphical meaning of tangent in optimization problem refers to the point where the slope of the tangent line is equal to the slope of the curve at a specific point. This point represents the optimal solution of the problem, as it is the point where the curve is changing direction and the slope is equal to the slope of the tangent line.

2. How is the tangent line used in optimization problems?

The tangent line is used in optimization problems to determine the optimal solution. By finding the point where the slope of the tangent line is equal to the slope of the curve, we can identify the point of maximum or minimum value, depending on the problem.

3. Can the tangent line intersect the curve at multiple points?

Yes, the tangent line can intersect the curve at multiple points. In fact, in some cases, there may be multiple tangent lines that intersect the curve at the same point. This often occurs in non-linear curves and can be useful in identifying multiple optimal solutions.

4. How does the slope of the tangent line relate to the rate of change?

The slope of the tangent line can be thought of as the instantaneous rate of change at a specific point on the curve. This means that the slope of the tangent line can help us understand how the curve is changing at that particular point, which is useful in optimization problems where we are looking for the maximum or minimum rate of change.

5. Is the tangent line always horizontal at the point of maximum or minimum value?

No, the tangent line is not always horizontal at the point of maximum or minimum value. In some cases, it may be vertical or have a different slope. This depends on the shape of the curve and the specific problem being solved. However, the key point is that the slope of the tangent line is equal to the slope of the curve at that point, regardless of its orientation.

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