Here’s the hard part. We always want the problem structured in a particular way. Here, we are choosing to maximize f (x, y) by choice of x and y .
The function g(x,y) represents a restriction or series of restrictions on our possible actions.
The setup for this problem is written as l(x,y)= f(x,y)+λg(x,y)
For example, a common economic problem is the consumer choice decision. Households are selecting consumption of various goods. However, consumers are not allowed to spend more than their income (otherwise they would buy infinite amounts of everything!!). Let’s set up the consumer’s problem:
Suppose that consumers are choosing between Apples (A) and Bananas (B). We have a utility function that describes levels of utility for every combination of Apples and Bananas.
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A 2 B 2 = Well being from consuming (A) Apples and (B) Bananas.
Next we need a set pf prices. Suppose that Apples cost $4 apiece and Bananas cost $2 apiece. Further, assume that this consumer has $120 available to spend. They the income constraint is
$2B+$4A≤$120
However, they problem requires that the constraint be in the form g(x, y)≥ 0. In
the above expression, subtract $2B and $4A from both sides. Now we have 0≤$120−$2B−$4A
g(A, B) Now, we can write out the lagrangian
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l(A,B)= A2 B2 +λ(120−2B−4A)
f (A, B) g(A, B)
Step II: Take the partial derivative with respect to each variable
We have a function of two variables that we wish to maximize. Therefore, there will be two first order conditions (two partial derivatives that are set equal to zero).
In this case, our function is
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l(A,B)= A2 B2 +λ(120−2B−4A)
Take the derivative with respect to A (treating B as a constant) and then take the derivative with respect to B (treating A as a constant).
Answer:
10.29
Step-by-step explanation:
use Pythagorean theorem
side opposite to 90 is always hypotenuse
and we know
h^2=p^2+b^2
you can put any side as perpendicular or base.
Answer:
The point where all three lines meet is (8, 11)
Step-by-step explanation:
The given functions are;
f(x) = x²⁽ˣ ⁻ ⁵⁾ + 3
g(x) = 2·x - 5
h(x) = 8/x + 10
Equating the simpler functions g(x) to h(x) to find the points of the same value gives;
2·x - 5 = 8/x + 10
2·x - 15 = 8/x
2·x² - 15·x = 8
2·x² - 15·x - 8 = 0
With the aid of an online application, we have;
(2·x + 1)(x - 8) = 0
x = -1/2, or x = 8
The y-values at the point of intersection, is given as follows;
g(x) = 2·x - 5 = 2×(-1/2) - 5 = -6 or g(x) = 2·x - 5 = 2×(8) - 5 = 11
The points where g(x) and h(x) meet are (-1/2, -6) and (8, 11)
To check where the point of intersection of the two functions g(x) ang h(x) meet f(x), we have;
At x = -1/2, f(x) = 
= 3.0442
At x = -1/2, f(x) = 2⁽⁸⁻⁵⁾ + 3 = 8 + 3 = 11
Therefore, the point where all three lines meet = (8, 11).
Answer:
c. 3x + 15
The son is currently 12 years old.
The father is currently 42 years old.
Step-by-step explanation:
x = the son's age 2 years ago
x + 2 = the son's age in the present
x + 5 = the son's age 3 years from now
f = the father's age 2 years ago = 4x
f + 2 = the father's age in the present = 4x + 2
f + 5 = the father's age 3 years from now = 3(x + 5) = 4x + 5
*3(x + 5) when distributed is equivalent to 3x + 15
4x + 5 = 3x + 15
x + 5 = 15
x = 10
This means that:
Present age of the son: (10) + 2 = 12 years old
Present age of the father: 4(10) + 2 = 42 years old
~Hope this Helps!~
