Question

Use Logrange method to
compute extreme values of
f(Ly,z) = xyz subject to the
constraint +9y+z=4.
Assume that uso for this
problem. Why is this assumpt
needed​

Answer 1

Answer:

Yes, your reasoning is mostly correct. You are solving the optimization problem by substitution (in comment told by Lagrange multiplier's method) with x,y,z,a∈R. At the end you have two cases: a>0 and a<0. If:

1) a>0⇒fxx=fyy=−2a3<0,Δ=a23>0⇒f(a3,a3,a3) is max.

2) a<0⇒fxx=fyy=−2a3>0,Δ=a23>0⇒f(a3,a3,a3) is min.

But remember that these are just local (not global) extremes.

To simplify the analysis, consider finding extreme values of y=3x−x3.

y′=3−3x2=0⇒x1,2=±1.

y′′=−6xy′′(−1)=6>0⇒y(−1)=−2 (local min)y′′(1)=−6<0⇒y(1)=2 (local max).

Global max/min of this function do not exist, because:

limx→±∞(3x−x3)=∓∞.