Answer:
Explanation:
Here, we want to get the maximum load supported
From the first part of the question, we have it that:
The maximum load g, varies jointly with the width w and the height h, and inversely as the length l between the supports
Mathematically, we have that as:
where k is the proportionality constant
Now, let us get the value of k
We substitute the first set of values as follows:
Now, using this value of k, we want to get the value of g
Mathematically, we have that as:
Looks like .
The latter two critical points occur outside of since
so we ignore those points.
The Hessian matrix for this function is
The value of its determinant at (0, 0) is , which means a minimum occurs at the point, and we have
.
Now consider each boundary:
which has 3 extreme values over the interval of 31/4 = 7.75 at the point (1, 1/2); 8 at (1, 1); and 10 at (1, -1).
and we get the same extrema as in the previous case: 8 at (-1, 1), and 10 at (-1, -1).
which doesn't tell us about anything we don't already know (namely that 8 is an extreme value).
which has 3 extreme values, but the previous cases already include them.
Hence has absolute maxima of 10 at the points (1, -1) and (-1, -1) and an absolute minimum of 0 at (0, 0).