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What are optimization problems?
Problems that involve finding the best possible solution, often maximizing or minimizing a certain quantity.
What does 'optimizing' mean in calculus?
Finding the best possible solution, often looking to maximize or minimize a certain quantity.
How are critical points used in optimization?
Critical points are candidates for maximum or minimum values of a function, found where the derivative is zero or undefined.
Steps to solve optimization problems?
- Identify the quantity to optimize. 2. Establish a function. 3. Reduce variables using constraints. 4. Find critical points. 5. Verify minimum or maximum.
How to deal with multiple variables?
Use given relationships to rewrite the equation in terms of a single variable through substitution.
How to find critical points?
Take the first derivative of the function, set it equal to zero, and solve for the variable(s).
How to verify if a critical point is a minimum?
Use the Second Derivative Test: if the second derivative is positive at the critical point, it's a minimum.
How to verify if a critical point is a maximum?
Use the Second Derivative Test: if the second derivative is negative at the critical point, it's a maximum.
What is the Second Derivative Test?
If (f'(c) = 0) and (f''(c) > 0), then (f) has a local minimum at (c). If (f'(c) = 0) and (f''(c) < 0), then (f) has a local maximum at (c).