286 PROBLEM OF MAXIMUM OR MINIMUM AREA. 



Example (5). Determine a point in a given curve so that 

 the area of the triangle formed by the tangent at that point 

 and the co-ordinate axes may be a maximum or a minimum. 



By Art. 260, the area varies as the product of 



dx dy 



x v-r , and u x-^-: 





dx 



then we require the maximum or minimum value of u, 

 It will be found that 



/ dy\ dy \d*y 

 (y-x-j'-} [x-f- 4- y j^ 

 V 7 dx) \ dx y J dj? 



du 

 dx 



v 

 Now, as we shall see in Chapter XXI., where -y^ = 0, 



the curve has in general a singular point called a point of 



inflection. Where y x -j- = 0, the tangent passes through 

 dx 



the origin and the area in question vanishes. It will be often 

 obvious when any particular curve is considered, that nei- 

 ther of these exceptional cases can hold. We have then 



(ill 



&-f-+y = Q as the condition which determines the point 

 required. 



When x -jt +y = 0, we have, by Art. 260, 



x = 2x, and y = 2y. 



Hence in general when the area is a maximum or a mini- 

 mum the portion of the tangent between the axes is bisected 

 at the point of contact. It will in general be obvious from 

 the figure in the case of any particular curve whether the 

 area is a maximum or minimum. 



