Draw a median from the vertex angle to the base.
(It is also the perpendiculat bsector of the base as
well as the bisector of the vertex angle).

That divides the isosceles triangle into two
congruent right triangles.

Let the height be x.  Then use the Pythagorean
theorem to find each half of the base:  .





Let the area be y.

 



Square both sides:





Find  implicitly:



Divide through by 2



Divide both sides by y



To find the maximum value set the derivative = 0



Multiply both sides by y



Divide through by 2





;  

           
             
          

Ignore the negative.

[The value x=0 gives the minimum value for the area, 0.]

The value  gives the maximum value.

To find what that maximum area is, substitute in











So the maximum area is 2.  {That is when the isosceles 

triangle is a right triangle, with vertex angle = 90°].

Edwin