📘 multivariable calculus

1. **Problem statement:** Analyze the function $$f(a,b) = (2a + 2b)^2 + (2a - 2b)^2$$ over variables $a$ and $b$. 2. **Simplify the function:**

1. Problem 17: Find the maximum and minimum values of the function $f(x,y)=x^3+3xy^2-15x^2-15y^2+72x$. 2. Compute partial derivatives and critical points.

1. Problem 17: Find and classify critical points of $f(x,y)=x^3+3xy^2-15x^2-15y^2+72x$. 2. Compute partial derivatives and set gradient to zero.

**Problem:** Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for the function $f(x,y) = e^{xy} \sin(4y^2)$. 1. **Write down the fun

1. **Given:** $f(x,y) = x^2 + xy^3$ a. $f(0,0) = 0^2 + 0\cdot0^3 = 0$

1. Find the function values for \( f(x,y) = x^2 + xy^3 \). 1.a. Calculate \( f(0,0) = 0^2 + 0 \times 0^3 = 0 \).

1. **Problem 1:** Given $z=x^3 + y^3 - 3x^2y$, verify $x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2

1. **State the problem:** We have a function of two variables: $$u(x_1, x_2) = 2 x_2$$.