Question

 4. In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle e is amazingly consistent. Based

 on the geometry of the cell, it can be shown that the surface area S is given by

 where s, the lengths of the sides of the hexagon, and h, the height, are constant. (See the figure on the next page.)

 a. Calculate dSIdθ.

 b. What angle should the bees prefer?

 c. Determine the minimum surface area of the cell (in terms of s and h).

 Note. Actual measurements of the angle e in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 2°.

 In problems (8) through (15) below, use complete sentences in your answers. You may use illustrations (examples or graphs) to assist you in your explanation; however, whether you use illustrations or not, you must have an reasonable explanation consisting of complete sentences.

 8. Consider the Extreme Value Theorem:

 If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d.

 Explain why the Extreme Value Theorem is true.

 where s, the lengths of the sides of the hexagon, and h, the height, are constant. (See the figure on the next page.)

 9. Consider Fermar's Theorem:

 If f has a local maximum or minimum at c (where c is not at an endpoint of the domain of f). and if f"(c) exists, then f'(c) = 0.

 Explain why the Fermat's Theorem is true.

 10. Recall the Closed Interval Method:

 To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b):

 a. Find the values of f at the critical numbers of f in (a, b).

 b. Find the values of f at the endpoints of the interval.

 c. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

 Explain why this three-step procedure always works.

 11. Some functions may not have a local minimum or maximum at x = c even though f"(c) = 0. Explain why.

 12. Some functions may have a local minimum or maximum at x = c even though f"(c) does not exist. Explain why. (Assume that x = c is in the domain of f.)

 13. Consider the First Derivative Test:

 a. If f' changes from positive to negative at c, then f has a local maximum at c.

 b. If f' changes from negative to positive at e, then f has a local minimum at c.

 e. If f' does not change sign at c (for example, if f' is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c.

 Explain why the First Derivative Test works.

 14. Consider the Second Derivative Test:

 Suppose f" is continuous near c. Then,

 a. If f'(c) = 0 and f"(c) >0, then f has a local minimum at c.

 b. If f'(c) = 0 and f"(c) <0, then f has a local maximum at c.

 Explain why the Second Derivative Test works.

 15. Suppose (c, f(c)) is an inflection point of y = f(x). Explain why f" changes signs at x=c.

 

Answer 1

by the HomeworkLib policy we can anser first full question please post again the remanng 8-15 question