Graphical meaning of tangent in optimization problem

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1. Jun 9, 2015

nigels

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.

2. Jun 9, 2015

Svein

Are you sure that you have given all information? The curve looks suspiciously like y = x2.

3. Jun 9, 2015

nigels

@Svein: Sorry, x1 and x2 mean x_1 and x_2.

4. Jun 10, 2015

Svein

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. Jun 10, 2015

theodoros.mihos

I think you need this