To maximize utility, your mother should purchase the flowers that provide the most satisfaction per dollar. Let x be the number of flower a and y be the number of flower b that your mother should purchase.
The total amount spent on flowers can be represented by the inequality: 2x + 3y ≤ 17, since she can spend at most $17 on flowers.
The utility function can be represented by: U(x,y) = x * y, since the utility of the bouquet is proportional to the product of the number of flowers of each type.
To maximize utility, we need to find the values of x and y that satisfy the inequality 2x + 3y ≤ 17 and maximize the function U(x,y) = x * y.
We can use the method of Lagrange multipliers to find the values of x and y that maximize the function U(x,y) subject to the constraint 2x + 3y ≤ 17.
The Lagrange function is: L(x,y,λ) = x * y + λ(17 - 2x - 3y)
Taking the partial derivatives of L with respect to x, y, and λ, and setting them to zero, we get:
∂L/∂x = y - 2λ = 0
∂L/∂y = x - 3λ = 0
∂L/∂λ = 17 - 2x - 3y = 0
Solving these equations simultaneously, we get:
y = 2λ
x = 3λ
17 - 2x - 3y = 0
Substituting y = 2λ and x = 3λ in the third equation, we get:
17 - 6λ - 6λ = 0
λ = 17/12
Substituting λ = 17/12 in the equations for x and y, we get:
x = 3λ = 17/4
y = 2λ = 17/6
Since we can only purchase whole numbers of flowers, the optimal number of flower a to purchase is 4, and the optimal number of flower b to purchase is 2. This will cost exactly $17 and will maximize the utility of the bouquet.