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Explain the role of derivatives in optimization.
Derivatives help find the minimum or maximum value of a function by identifying critical points where the rate of change is zero.
Explain the Second Derivative Test in optimization.
The Second Derivative Test determines concavity at critical points; positive concavity indicates a minimum, negative indicates a maximum.
Why is substitution important in optimization problems?
Substitution reduces the number of variables, allowing differentiation with respect to a single variable.
What is the goal of optimization problems?
The goal is to find the maximum or minimum value of a function subject to given constraints.
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).
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.