15 Questions
If a function is concave up on an interval, what kind of point can it have in that interval?
Local minimum
What is the significance of a critical number of a function?
It could be a possible location for a local maximum, local minimum, or an inflection point.
How does changing the domain of a function affect its function values?
It can change the range of the function but may not affect the function's critical points.
What can be said about the absolute maximum and minimum values of a function on a closed interval?
They occur at critical points or at the endpoints of the interval.
How can inflection points be identified on a graph?
Inflection points occur where the concavity of the function changes.
What is the purpose of finding critical numbers of a function?
To locate potential locations of local extrema or inflection points.
Provide an example of a function f where at least one of the hypotheses of Rolle's theorem is not satisfied on the interval [a, b], but f has a horizontal tangent at x = c where c is in (a, b).
f(x) = x^3 - 3x^2 + 2x
Give an example of a function g defined on [-4, 4] that meets the following criteria: no absolute maximum, exactly one local maximum, an absolute minimum, no local minimum, and exactly 2 critical values.
g(x) = x^3 - 3x^2 + 2x
Explain how changing the domain of a function can affect the function's values.
Changing the domain can restrict the input values, causing the function's output values to change or become undefined for certain inputs.
What is an inflection point on a graph? Provide a brief explanation.
An inflection point on a graph is where the concavity changes from positive to negative or vice versa. It is where the curve transitions from being concave up to concave down, or the reverse.
Define what an absolute maximum and an absolute minimum are for a function.
An absolute maximum is the highest value that a function reaches over its entire domain. An absolute minimum is the lowest value that a function reaches over its entire domain.
What are critical numbers of a function? Provide a simple explanation.
Critical numbers of a function are the points where the derivative is either zero or undefined. They are potential locations for extrema or points of inflection.
State one method to find the temperature at which the engine is heating up the fastest.
Maximize the derivative of T
If f has only a local minimum at x=1, what type of extrema does g(x) have on [1,5]?
g(x) has a local maximum at x=1
Which statement among the given choices must be false?
If f'(x) < 0 for all x-values, then f(5) ≠ f(-2)
Test your knowledge on mathematical statements and functions. Circle all the statements that must be true based on the given options and graph representations.
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