We are given the equation: g(s,t) = t^2 e^s
Which is subject to the constraint: s^2 + t^2=3
The points (x,y) that will maximize g(s,t) will be the
points that will satisfy the equation ∇f(x, y, z)
= λ∇g(x, y, z), therefore:
2 t e^s = λ (2 t), so e^s = λ --> 1
t^2 e^s = λ (2 s) --> 2
s^2 + t^2 = 3 -->3
To solve this problem, note that λ cannot be
zero by equation 1 since e^s can never be zero. Therefore plug in equation 1 to
2:
t^2 e^s = (e^s) (2 s)
t^2 = 2 s -->
4
Plug in equation 4 to 3:
s^2 + 2s = 3
By completing the square:
s^2 + 2s + 1 = 4
(s + 1)^2 = 4
s + 1 = ±2
s = -3, 1
Calculating for t using equation 4:
when s = -3
t^2 = 2(-3)
t = sqrt(-6)
Since t is imaginary, therefore s=-3 is not a solution
when s = 1
t^2 = 2(1)
t = sqrt(2) = ±1.414
Therefore the maxima and minima points are at:
(1, -1.414) and (1, 1.414)
g(1, -1.414)=(-1.414)^2 e^(1) = 5.434
g(1, -1.414)=( 1.414)^2 e^(1) = 5.434
