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What are absolute extrema?
The maximum and minimum values of a function over its entire domain.
What are critical points?
Points where the first derivative of the function is equal to zero or undefined.
What is a closed interval?
An interval that includes its endpoints.
Define absolute maximum.
The largest y-value of a function on a given interval.
Define absolute minimum.
The smallest y-value of a function on a given interval.
What is the domain of a function?
The set of all possible input values (x-values) for which the function is defined.
What is a candidate in the Candidates Test?
A critical point or endpoint of a function on a closed interval that is evaluated to find absolute extrema.
What does 'evaluate a function' mean?
To find the value of the function at a specific input (x-value).
What is the first derivative of a function?
The rate of change of the function with respect to its input variable.
What is the range of a function?
The set of all possible output values (y-values) of the function.
How do you find critical points of a function?
1. Find the first derivative. 2. Set the derivative equal to zero and solve for x. 3. Find where the derivative is undefined.
What are the steps to apply the Candidates Test?
1. Find critical points. 2. Evaluate the function at critical points. 3. Evaluate the function at endpoints. 4. Compare y-values.
How do you determine the absolute maximum using the Candidates Test?
After evaluating the function at critical points and endpoints, the largest y-value is the absolute maximum.
How do you determine the absolute minimum using the Candidates Test?
After evaluating the function at critical points and endpoints, the smallest y-value is the absolute minimum.
How do you handle critical points outside the given interval?
Discard any critical points that do not fall within the specified closed interval.
How do you find the derivative of a polynomial function?
Apply the power rule: if $f(x) = ax^n$, then $f'(x) = nax^{n-1}$.
How do you factor a quadratic equation?
Find two numbers that multiply to the constant term and add up to the coefficient of the linear term.
How do you evaluate a function at a specific point?
Substitute the given x-value into the function and simplify to find the corresponding y-value.
How do you simplify an algebraic expression?
Combine like terms, distribute constants, and apply algebraic rules to reduce the expression to its simplest form.
How do you solve for x when the derivative is zero?
Set the derivative equal to zero and use algebraic techniques such as factoring, quadratic formula, or other methods to find the values of x.
Explain the Candidates Test.
A method to find absolute extrema by evaluating a function at critical points and endpoints.
Why are critical points important for finding absolute extrema?
Absolute extrema can only occur at critical points or endpoints on a closed interval.
How do you determine if a critical point is a local or absolute extremum?
Compare the function value at the critical point with the function values at the endpoints of the interval.
What is the significance of a closed interval when finding absolute extrema?
A closed interval guarantees the existence of absolute extrema if the function is continuous on that interval.
Describe the relationship between derivatives and absolute extrema.
Derivatives are used to find critical points, which are potential locations of absolute extrema.
Why do we evaluate endpoints in the Candidates Test?
Absolute extrema can occur at the endpoints of the closed interval.
What does it mean for a function to be continuous?
A function is continuous if it has no breaks, jumps, or holes in its graph.
Explain the relationship between local and absolute extrema.
A local extremum is a maximum or minimum within a specific neighborhood, while an absolute extremum is the overall maximum or minimum on the entire domain or a given interval.
How does the Candidates Test simplify finding absolute extrema?
It narrows down the possible locations of absolute extrema to critical points and endpoints, making the search more efficient.
Describe the role of the first derivative in the Candidates Test.
The first derivative is used to find critical points by setting it equal to zero or identifying where it is undefined.