Graphical meaning of tangent in optimization problem

[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.

 

[Svein]

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

 

[nigels]

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

 

[Svein]

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.

 

[theodoros.mihos]

I think you need this