Consider the Lagrange problem:
 

Use the envelope theorem to estimate the maximum value  when , and verify this by computing the optimal value function .
b) Maximize the function
 

with respect to .

### Part (a): Envelope Theorem Application for the Lagrange Problem

We are given the Lagrange problem:

- Objective function: \( f(u, v) = u^3 + v \)
- Constraint: \( g(u, v) = u^2 + v^2 = a^2 \)

We aim to use the envelope theorem to estimate the maximum value of \( f(a) \) when \( a = 1.01 \) and verify this by computing the optimal value function \( f(a) \).

#### Step 1: Set up the Lagrangian

We set up the Lagrangian function by introducing a Lagrange multiplier \( \lambda \) for the constraint:

\[
\mathcal{L}(u, v, \lambda) = u^3 + v + \lambda (a^2 - u^2 - v^2)
\]

#### Step 2: First-Order Conditions

To find the optimal values of \( u \) and \( v \), we take the partial derivatives of the Lagrangian with respect to \( u \), \( v \), and \( \lambda \), and set them equal to zero.

- Derivative with respect to \( u \):

\[
\frac{\partial \mathcal{L}}{\partial u} = 3u^2 - 2\lambda u = 0
\]
\[
u(3u - 2\lambda) = 0
\]
This gives two possibilities: either \( u = 0 \) or \( \lambda = \frac{3u}{2} \).

- Derivative with respect to \( v \):

\[
\frac{\partial \mathcal{L}}{\partial v} = 1 - 2\lambda v = 0
\]
\[
v = \frac{1}{2\lambda}
\]

- Derivative with respect to \( \lambda \) (the constraint):

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