Answer:
a) The function is: f(x, y) = x + y.
The constraint is: x*y = 196.
Remember that we must write the constraint as:
g(x, y) = x*y - 196 = 0
Then we have:
L(x, y, λ) = f(x, y) + λ*g(x, y)
L(x, y, λ) = x + y + λ*(x*y - 196)
Now, let's compute the partial derivations, those must be zero.
dL/dx = λ*y + 1
dL/dy = λ*x + 1
dL/dλ = (x*y - 196)
Those must be equal to zero, then we have a system of equations:
λ*y + 1 = 0
λ*x + 1 = 0
(x*y - 196) = 0
Let's solve this, in the first equation we can isolate λ to get:
λ = -1/y
Now we can replace this in the second equation and get;
-x/y + 1 = 0
Now let's isolate x.
x = y
Now we can replace this in the last equation, and we will get:
(x*x - 196) = 0
x^2 = 196
x = √196 = 14
then the minimum will be:
x + y = x + x = 14 + 14 = 28.
b) Now we have:
f(x) = x*y
g(x) = x + y - 196
Let's do the same as before:
L(x, y, λ) = f(x, y) + λ*g(x, y)
L(x, y, λ) = x*y + λ*(x + y - 196)
Now let's do the derivations:
dL/dx = y + λ
dL/dy = x + λ
dL/dλ = x + y - 196
Now we have the system of equations:
y + λ = 0
x + λ = 0
x + y - 196 = 0
To solve it, we can isolate lambda in the first equation to get:
λ = -y
Now we can replace this in the second equation:
x - y = 0
Now we can isolate x:
x = y
now we can replace that in the last equation
y + y - 196 = 0
2*y - 196 = 0
2*y = 196
y = 196/2 = 98
The maximum will be:
x*y = y*y = 98*98 = 9,604
