The problem is asking us to find the extreme values of the function:

$f(x, y) = 2x^2 + 3y^2 - 4x - 5$

subject to the constraint:

$x^2 + y^2 \leq 16.$

Steps to Solve:

1. Find the critical points of $f(x, y)$:

   • Compute the partial derivatives of $f(x, y)$: $f_x = \frac{\partial}{\partial x}(2x^2 + 3y^2 - 4x - 5) = 4x - 4,$ $f_y = \frac{\partial}{\partial y}(2x^2 + 3y^2 - 4x - 5) = 6y.$
   • Set $f_x = 0$ and $f_y = 0$ to find critical points: $4x - 4 = 0 \implies x = 1,$ $6y = 0 \implies y = 0.$ Thus, the critical point is $(1, 0)$.

2. Check the boundary $x^2 + y^2 = 16$:

   • On the boundary, we use the constraint $x^2 + y^2 = 16$ and substitute $y$ in terms of $x$ or vice versa.
   • Rewrite $f(x, y)$ with the constraint and analyze its behavior using Lagrange Multipliers.

3. Apply Lagrange Multipliers:

   • Define $g(x, y) = x^2 + y^2 - 16 = 0$.

   • Solve $\nabla f = \lambda \nabla g$, where: $\nabla f = (4x - 4, 6y), \quad \nabla g = (2x, 2y).$ This gives the system: $4x - 4 = \lambda (2x),$ $6y = \lambda (2y),$ $x^2 + y^2 = 16.$

     • From $6y = \lambda (2y)$, if $y \neq 0$, $\lambda = 3$.
     • From $4x - 4 = \lambda (2x)$, $4x - 4 = 6x$, so $x = -2$.

     Substituting $x = -2$ into $x^2 + y^2 = 16$: $(-2)^2 + y^2 = 16 \implies 4 + y^2 = 16 \implies y^2 = 12 \implies y = \pm 2\sqrt{3}.$

     Additional points to evaluate are: $(x, y) = (1, 0), (-2, 2\sqrt{3}), (-2, -2\sqrt{3}).$

4. Evaluate $f(x, y)$ at all critical points and boundary points:

   • At $(1, 0)$: $f(1, 0) = 2(1)^2 + 3(0)^2 - 4(1) - 5 = 2 - 4 - 5 = -7.$
   • At $(-2, 2\sqrt{3})$: $f(-2, 2\sqrt{3}) = 2(-2)^2 + 3(2\sqrt{3})^2 - 4(-2) - 5,$ $= 2(4) + 3(12) + 8 - 5 = 8 + 36 + 8 - 5 = 47.$
   • At $(-2, -2\sqrt{3})$: $f(-2, -2\sqrt{3}) = 47 \quad \text{(same as above)}.$

5. Determine the extreme values:

   • Minimum value: $f(1, 0) = -7$,
   • Maximum value: $f(-2, \pm 2\sqrt{3}) = 47$.

Final Answer:

• The maximum value is $47$.
• The minimum value is $-7$.

Do you want a step-by-step explanation of any part? Here are five related questions to explore further:

1. How do Lagrange Multipliers work, and why do we use them for constraints?
2. Can we solve this problem graphically for better understanding?
3. How do we verify if critical points are maxima, minima, or saddle points?
4. What happens if the constraint is an inequality rather than equality?
5. How does the method extend to three or more variables?

Tip: Always check both the interior critical points and the boundary when solving constrained optimization problems!