maxpywx

(x,y)R2+

subjectto 4ln(x+1)=y.

a. On thex-yplane, given anyx0, the slope of the graph of the constraint 4 ln(x+1) =yis equal to (the answer should be an expression ofx).

b. Supposethatp=10andw=6.

1. On thex-yplane, given anyx0, the slope of the level surface of the objective function 10y6xis equal to
2. The optimum (x,y) is:x= andy=
3. At this optimum, the gradient of the objective 10y6xis equal to
4. At this optimum, which of the following vectors is aligned with the gradient of the

constraint function 4 ln(x+ 1)y?

10

A.

6

6

B.

10

6

C.

10

6/10

D.

1

E. none of the above

c. Suppose thatpremains to be 10 whilewvaries.

i. The optimum is uniquely (0,0) if and only ifwis

than, equal to [fill in the blank with one of the three items]) number]

(higher than, lower [provide a

ii. The equation system produced by the Lagrange method with equality constraints is valid whenwis (higher than, lower than [fill in the blank with one of the two items]) [provide a number]