1. **Stating the problem:** Optimization problems involve finding the maximum or minimum values of a function, often subject to certain constraints. 2. **Key formulas and concepts:** - Objective function: The function $f(x)$ or $f(x,y)$ to be maximized or minimized. - Critical points: Points where the derivative(s) equal zero, i.e., $f'(x) = 0$ or $\nabla f(x,y) = \mathbf{0}$. - Second derivative test (1D): If $f''(x) > 0$, local minimum; if $f''(x) < 0$, local maximum. - Hessian matrix (multivariable): $H = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}$. - Positive definite Hessian implies local minimum; negative definite implies local maximum. 3. **Important rules:** - Find critical points by setting derivatives equal to zero. - Use second derivative or Hessian to classify critical points. - Check boundary or constraint conditions if applicable. 4. **Common optimization formulas:** - For single variable: $$\text{Find } x \text{ such that } f'(x) = 0$$ - For multivariable: $$\text{Find } (x,y) \text{ such that } \frac{\partial f}{\partial x} = 0, \quad \frac{\partial f}{\partial y} = 0$$ - Lagrange multipliers for constrained optimization: $$\nabla f = \lambda \nabla g$$ where $g(x,y) = 0$ is the constraint. 5. **Summary:** - Set derivatives to zero to find candidates. - Use second derivative or Hessian to determine maxima or minima. - Consider constraints with Lagrange multipliers. This collection covers the main formulas and rules used in optimization problems.