1. Let's state the problem: Optimization involves finding the maximum or minimum value of a function, often subject to certain constraints. 2. The general approach uses the derivative: To find extrema (maxima or minima), we find where the derivative of the function equals zero, i.e., solve $f'(x) = 0$. 3. Important rules: - Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined. - Use the second derivative test: if $f''(x) > 0$, the point is a local minimum; if $f''(x) < 0$, it is a local maximum. 4. Steps to solve an optimization problem: - Define the function to optimize. - Find its derivative. - Solve $f'(x) = 0$ to find critical points. - Use the second derivative test or analyze endpoints to determine maxima or minima. 5. Example: Suppose we want to maximize $f(x) = -2x^2 + 4x + 1$. - Derivative: $f'(x) = -4x + 4$. - Set derivative to zero: $-4x + 4 = 0 \Rightarrow x = 1$. - Second derivative: $f''(x) = -4 < 0$, so $x=1$ is a maximum. - Maximum value: $f(1) = -2(1)^2 + 4(1) + 1 = -2 + 4 + 1 = 3$. This is the basic method for optimization problems.