. Folding a Pyramid - A pyramid with a square base and four faces,

each in the shape of an isosceles triangle, is made by cutting away four
triangles from a square piece of cardboard and bending up the resulting
triangles to form the walls of the pyramid. What is the largest volume
the pyramid can have assuming that the square piece of cardboard has
sides measuring a m?

I got everything but the I can't see to find the right solutions, I need help with my differentiation

To find the largest volume of the pyramid, we need to optimize the volume function with respect to the variable 'a' (side length of the square cardboard).

Let's start by setting up the problem.

1. Determine the height of the pyramid:
- The height can be found as the perpendicular distance from the base of the pyramid to its apex.
- Since the pyramid is formed by bending up the triangular faces, the height can be found by cutting one of the triangular faces into two congruent right triangles.
- The height of the pyramid will then be the hypotenuse of these right triangles.

2. Express the volume of the pyramid in terms of 'a' and 'h':
- The volume of a pyramid can be calculated using the formula V = (1/3) * base area * height.
- The base area is the square of side length 'a'.
- The height, 'h', is already determined in step 1.

3. Optimize the volume function:
- Differentiate the volume function with respect to 'a' and set the derivative equal to zero to find possible critical points.
- Check the critical points using the first derivative test or second derivative test.
- Determine whether each critical point is the maximum or minimum point.
- If all critical points are evaluated and there are no local maximums, consider the endpoints of the possible range.

By following these steps, you should be able to find the largest volume that the pyramid can have with the given square cardboard.