Question

Anybody please

Use the method of Lagrange multipliers to find the max and min values
of g(s,t)= t^2 e^s given that s^2+t^2=3 . How are we assured that these
extreme values exist?

I am having Difficulty solving the system of lagrange equation

Thank you.

Answer 1

2λs=t^2e^s
2λt=2te^s

s^2+t^2=3

Answer 2

Thanks alot.

Answer 3

just ripping mechanically through the problem gives you something like this


\(\large g(s,t)= e^s t^2, \qquad h(s,t) = s^2+t^2=3\)

\(\large \nabla g = \lambda \nabla h \) [the central idea]
\(\large \implies <e^s t^2, e^s 2t> = \lambda <2s, 2t>\)
\(\large \implies \dfrac{e^st^2}{2s} = \dfrac{e^s2t}{2t}\)
\(\large \implies 2s = t^2\)


and from \(\large h(s,t) = s^2+t^2=3\) you have \(\large s = -3, 1\) and \(t = \dots\)

Answer 4

if you are interested, we can work through what the problem actually explores, in termsof the geometry/physical interpretation.

Answer 5

Sorry how did you get s=-3 isnt suppose to be 3

Answer 6

s^2+2s=3

s(s+2)=3

s=3 s=-2

Answer 7

\(s^2 + 2s = 3\)
\(s^2 + 2s - 3 = 0\)
\((s+ 3)(s-1) = 0\)
\(s = -3, 1\)

Answer 8

And no t value for s=-3

Answer 9

no. according to me.

can you see this in your head?

the underlying function is being constrained by/ looked at around a circle or radius \(\sqrt{3}\)

Answer 10

Thank you.

Answer 11

Ok, can you please work me through the questions in terms of the geometry/physical interpretation.