If a function has a local maximum at a point inside the interval, its derivative:

Could be either zero or undefined

The Extreme Value Theorem guarantees that:

A continuous function over a closed interval has both absolute maximum and minimum

There is always a critical point

Local maxima equal absolute maxima

The function is differentiable everywhere

If the interval is open or the function is discontinuous, then:

The derivative is always undefined

An endpoint extremum is:

Not considered a local extremum

To find absolute extrema:

b) Test all endpoints and critical points

a) Test only where derivative is positive

Both an absolute maximum and minimum

A function increasing throughout an interval:

Has no local maxima or minima inside

Must have derivative equal to zero somewhere

Must have local maxima and minima

Rolle’s theorem applies if a function is:

Continuous on [a, b] and differentiable on (a, b), with f(a) = f(b)

Continuous but not differentiable

Differentiable but not continuous

Which of these is not a condition of Rolle’s theorem?

Function must have a maximum at x=0

Function is continuous over [a, b]

Function is differentiable over (a, b)

The Mean Value Theorem generalizes Rolle’s theorem by:

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) equals the slope of the secant line from A to B

What does the sign of the first derivative (f′) tell us about the shape of a function's graph?

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 has a local minimum.

A point c is called a critical point of a function f if what condition is met?

f′(c)=0 or f′(c) is undefined

The function changes concavity at c .

(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) = 3x 4 + 8x 3 - 18x 2

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 = 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

Calculus: Extreme Values Quiz

Authored by Michael Sabugaa

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MULTIPLE CHOICE QUESTION

30 sec • 2 pts

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

MULTIPLE CHOICE QUESTION

30 sec • 2 pts

The Extreme Value Theorem applies only when the function is:

Continuous over an open interval

Continuous over a closed, bounded interval

Discontinuous

Undefined

MULTIPLE CHOICE QUESTION

30 sec • 2 pts

An absolute minimum can occur:

Only at endpoints

Only at critical points

At endpoints or critical points

Nowhere

MULTIPLE CHOICE QUESTION

30 sec • 2 pts

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

MULTIPLE CHOICE QUESTION

30 sec • 2 pts

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

MULTIPLE CHOICE QUESTION

30 sec • 2 pts

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

MULTIPLE CHOICE QUESTION

30 sec • 2 pts

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

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Calculus: Extreme Values Quiz

What is the absolute maximum of a function over an interval?

The Extreme Value Theorem applies only when the function is:

An absolute minimum can occur:

If a function is not continuous over a closed interval, it may:

A local maximum occurs at c if:

A critical point of f is where:

According to Fermat’s theorem, if ff has a local extremum at c and is differentiable there, then:

All critical points are:

To find absolute extrema on a closed interval, you evaluate:

If a function’s derivative does not exist at a point, this point:

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