Minimizing the Area of a Triangle

[b]Suppose a segment is drawn in the first quadrant of the coordinate plane and has variable slope. [/b][br][b]Suppose this segment[color=#9900ff] passes through the point (2,3). [/color][/b][color=#9900ff] [/color][br]This line also has an [b][color=#cc0000]x-intercept of (c,0)[/color][/b] and a [b]y-intercept of (0,d)[/b], where c, d > 0.  [br][br]Use calculus to algebraically determine the slope of this line for which the [b][color=#bf9000]area of the displayed right triangle is minimum. [/color][/b]Then use calculus to [i]prove[/i] this area is indeed the minimum area.[br][br][b]How does your result compare with what this applet suggests? [br]Note: [color=#cc0000]The red point is moveable. [/color][/b][br]
[b]Note to Instructors/Students:[br][/b][br][b][color=#9900ff]The purple point is moveable too, should you wish to solve a problem with similar context.[/color][/b]
Quick (Silent) Demo