Question

A rectangular area adjacent to a river is to be fenced in, but no fencing is required on the side by the river. The total area to be enclosed is 114,996 square feet. Fencing for the side parallel to the river is $6 per linear foot, and fencing for the other two sides is $7 per linear foot. The four corner posts cost $25 apiece. Let xx be the length of the one the sides perpendicular to the river.

[A] Find a cost equation C(x)C(x):
C(x)=C(x)=    

[B] Find C'(x)C′(x):
C'(x)=C′(x)=    

[C] Find the appropriate critical value(s) for the appropriate domain in the context of the problem.
   

[D] Perform the second derivative test to determine if there is an absoulte minimum at the critical value found.
C''(x)=C′′(x)=   

[E] What is the best conclusion regarding an absolute maximum or minimum at this critical value. (MULTIPLE CHOICE)

a) At the critical value C''(x)>0C′′(x)>0 so I can conclude that there is a local/relative maximum there but I can't conculde anything about an absolute maximum for x>0x>0

b) Since C''(x)>0C′′(x)>0 for all x>0x>0 we can colude that the CC is concave up for all values of x>0x>0 and that we therefore have an absolute minimum at the critical value for x>0x>0

c) Since C''(x)>0C′′(x)>0 for all x>0x>0 we can colude that the CC is concave up for all values of x>0x>0 and that we therefore have an absolute maximum at the critical value for x>0x>0

d) The second derivative test is inclusive with regards to an absolute maximum or minimum and the first derivative test should be performed

e) At the critical value C''(x)>0C′′(x)>0 so I can conclude that there is a local/relative minimum there but I can't conculde anything about an absolute minimum for x>0x>0

[F] Find the minimum cost to build the enclosure: $

*Please show all work associated*

Answer 1