WorksheetsCalculus: Extreme Values Quiz
Total questions: 43
Worksheet time: 40mins
What is the absolute maximum of a function over an interval?
The smallest value
The largest value
The value at a critical point
Always zero
The Extreme Value Theorem applies only when the function is:
Continuous over an open interval
Continuous over a closed, bounded interval
Discontinuous
Undefined
An absolute minimum can occur:
Only at endpoints
Only at critical points
At endpoints or critical points
Nowhere
If a function is not continuous over a closed interval, it may:
Still have absolute extrema
Not have absolute extrema
Always have a minimum
Always have a maximum
A local maximum occurs at c if:
f(c) ≥ f(x) for all x near c
f(c) ≤ f(x) for all x near c
f(c) is undefined
f'(c) is always positive
A critical point of f is where:
f′(c)>0
f′(c)<0
f′(c)=0 or f′(c)f'(c) is undefined
f(c)=0
According to Fermat’s theorem, if ff has a local extremum at c and is differentiable there, then:
f′(c)=0
f′(c)>0
f′(c)<0
f′(c) is undefined
All critical points are:
Local maxima
Local minima
Candidates for extrema
Absolute extrema
To find absolute extrema on a closed interval, you evaluate:
Only at endpoints
Only at critical points
Endpoints and critical points
At zero
If a function’s derivative does not exist at a point, this point:
Cannot be critical
Must be an endpoint
Can still be a critical point
Must be a maximum
If a function has a local maximum at a point inside the interval, its derivative:
Must be positive
Must be zero
Must not exist
Could be either zero or undefined
The Extreme Value Theorem guarantees that:
There is always a critical point
A continuous function over a closed interval has both absolute maximum and minimum
Local maxima equal absolute maxima
The function is differentiable everywhere
If the interval is open or the function is discontinuous, then:
Absolute extrema may not exist
Absolute extrema must exist
There are no critical points
The derivative is always undefined
Local extrema correspond to:
Only endpoints
Interior critical points
Always to f(0)
Non-differentiable points only
An endpoint extremum is:
A local extremum
Not considered a local extremum
Not an extremum
Always a critical point
To find absolute extrema:
a) Test only where derivative is positive
b) Test all endpoints and critical points
c) Test random points
d) Evaluate only at x = 0
A function may have:
Both an absolute maximum and minimum
Only one extremum
Neither maximum nor minimum
Any of the above
At a corner point of a graph
Derivative does not exist
Derivative exists
Function is discontinuous
There is no extremum
The largest and smallest values of a function are useful for
Making graphs only
Checking continuity
Finding intercepts
Solving optimization problems
A function increasing throughout an interval:
Cannot be continuous
Must have derivative equal to zero somewhere
Must have local maxima and minima
Has no local maxima or minima inside
Rolle’s theorem applies if a function is:
Continuous but not differentiable
Differentiable but not continuous
Continuous on [a, b] and differentiable on (a, b), with f(a) = f(b)
None of the above
In Rolle’s theorem, if f(a)=f(b), then there exists at least one cc such that:
f(c) > 0
f'(c) = 0
f'(c) > 0
f(c) = 0
Which of these is not a condition of Rolle’s theorem?
Function is continuous over [a, b]
Function is differentiable over (a, b)
f(a)=f(b)
Function must have a maximum at x=0
The Mean Value Theorem generalizes Rolle’s theorem by:
Allowing f(a)≠f(b)
Requiring f′(x)>0
Requiring the function to be undefined at some point
Removing the need for differentiability
According to the MVT, there exists a point C such that:
f(c)=0
f′(c)=0
f′(c) equals the slope of the secant line from A to B
f'(c) < 0
If a function is discontinuous on [a, b], the MVT:
Still applies
Does not apply
Guarantees f′(x)=0
Always produces f(a) = f(b)
The MVT can be used to show that if f'(x) > 0 on an interval, then f(x) is:
Constant
Decreasing
Increasing
Undefined
If the derivative of a function is zero over an interval, then the function is:
Increasing
Decreasing
Constant
Undefined
The MVT can also be used to demonstrate that two functions with the same derivative differ by:
Zero
A constant
Their endpoints
Nothing at all
When dropping a ball from a height, the MVT guarantees that at some point the instantaneous velocity equals:
Zero
The average velocity
Twice the average velocity
Maximum velocity
In the example of driving a car with average velocity 45 mph over an hour, the MVT ensures that at some point, the instantaneous velocity was:
45 mph
0 mph
Undefined
Undefined
In Rolle’s theorem, if f(x)f(x) is not differentiable at even one point in (a, b), then:
The theorem still applies
f(a)=f(b)
f(x) is always zero
The conclusion may not hold
If f'(x) > 0 on an interval, then f(x) is:
Constant
Decreasing
Increasing
Undefined
If f'(x) < 0 on an interval, then f(x) is:
Constant
Increasing
Decreasing
Undefined
Two differentiable functions that have the same derivative differ by:
A constant
Zero
Their maximum value
An endpoint
What does the sign of the first derivative (f′) tell us about the shape of a function's graph?
Concavity
Inflection points
Local extrema
Whether the function is increasing or decreasing
If the derivative of a function (f′) is positive over an interval, what can we conclude about the function over that interval?
The function is increasing.
The function is decreasing.
The function is concave down
The function has a local minimum.
What happens to a function f if its derivative (f′) is negative over an interval?
f has a local maximum.
f is concave up.
f is increasing.
f is decreasing.
A point c is called a critical point of a function f if what condition is met?
f(c)=0
f′(c)=0 or f′(c) is undefined
f′′(c)=0
The function changes concavity at c.
According to the First Derivative Test, if f′ changes sign from positive to negative at a critical point c, what does f(c) represent?
A local minimum
An inflection point
A local maximum
No local extremum
Find the critical points in the domains of the functions y = 4x3 – 3x
0
+/- 1
+/- 0.5
undefined
(5pts) Find the local and/or absolute maxima for the functions
over the specified domain [0, 2π] at f(x) = x + sinx
Absolute maximum: x = 2π, y = 2π; absolute minimum: x = 0, y = 0
Absolute maximum: x = 2π, y = 2π; absolute minimum: x = 0, y = 1
Absolute maximum: x = 2π, y = 2π; absolute minimum: x = -1, y = -1
Absolute maximum: x = 2π, y = 2π; absolute minimum: x = 1, y = 1
(10pts) Find the local and/or absolute minima and maxima for the functions
over (-∞, ∞) at f(x) = 3x4 + 8x3 - 18x2
Absolute minimum:
x = -3, y = -135; local maximum:
x = 1, y = 0; local minimum:
x = 1, y = 7
Absolute minimum:
x = -3, y = -135; local maximum:
x = 0, y = 1; local minimum:
x = 1, y = 7
Absolute minimum:
x = -3, y = -135; local maximum:
x = 0, y = 0; local minimum:
x = 1, y = 7
Absolute minimum:
x = -3, y = -135; local maximum:
x = 0, y = 0; local minimum:
x = 1, y = -7
