I am not sure what to do for this question
We'll have to do this without a graphing calculator. First, we need to find a function for the perimeter of the isosceles triangle. We know that the triangle has a fixed perimeter, so
Where x is the two equal sides, and y is the base (unequal) side. c is a constant. Now, we care about the area of the triangle. This can be calculated by
The base is going to be equal to y, and we can find the height using pythag.
So, our area function is going to be
We can rewrite our perimeter function like this
and sub every y for the function above.
Great! Now, we're looking for a maximum area. So, let's differentiate the function. Since you can use a graphic calculator, I'm going to use wolfram alpha.
Clearly, the turning point will occur when
I'll leave you to show that this is a maximum. Well, if two of the sides (ie. x) are equal to c/3, then the third side must be equal to c/3 (as the perimeter must add up to c). Therefore, the triangle is equilateral