Let's go through each question step-by-step to analyze and solve them.

Question 1

Problem: Locate all the relative maxima, relative minima, and saddle points for the function: $f(x, y) = x^2 + xy - 2y - 2x + 1.$

Solution Steps:

1. Compute the partial derivatives $f_x$ and $f_y$.
2. Set $f_x = 0$ and $f_y = 0$ to find the critical points.
3. Use the second derivative test to classify each critical point:
   • Compute $f_{xx}$, $f_{yy}$, and $f_{xy}$.
   • Determine the value of the discriminant $D = f_{xx}f_{yy} - (f_{xy})^2$ at each critical point.
   • Classify each point based on the value of $D$ and $f_{xx}$.

Question 2

Problem: Evaluate the iterated integrals:

1. $\int_{-1}^2 \int_2^4 (2x^2y + 3xy^2) \, dx \, dy$
2. $\int_0^1 \int_0^1 \frac{x}{(xy + 1)^2} \, dx \, dy$

Solution Steps:

1. Evaluate the inner integral for each iterated integral.
2. Substitute the result into the outer integral and evaluate it.

Question 3

Problem: Use a double integral to find the volume under the plane $z = 2x + y$ over the rectangle $R = (x, y) : 3 \leq x \leq 5, \, 1 \leq y \leq 2$.

Solution Steps:

1. Set up the double integral for the volume, which is given by: $\int_{3}^{5} \int_{1}^{2} (2x + y) \, dy \, dx$
2. Evaluate the inner integral with respect to $y$.
3. Evaluate the resulting outer integral with respect to $x$.

Would you like a detailed solution for each question? Here are some related questions you might consider:

1. How do you determine saddle points for a given function?
2. What is the significance of the discriminant in the second derivative test?
3. How can we simplify iterated integrals with complex functions in the integrand?
4. How does the region of integration affect the setup of a double integral for volume?
5. Are there alternative methods to evaluate double integrals in terms of polar coordinates?

Tip: For the second derivative test, remember that if the discriminant $D < 0$, it indicates a saddle point, where the function has neither a local max nor a min at that point.