Calculus Key Concepts: Critical Points, Extrema, and Concavity

Critical Number

A value c in the domain of f where f'(c)=0 or f'(c) does not exist.

When is c NOT a critical number?

If f(c) does not exist, then c is not a critical number.

Relative Maximum

A point where f(c) is greater than nearby function values.

Relative Minimum

A point where f(c) is less than nearby function values.

Absolute Maximum

The highest function value on an interval.

Absolute Minimum

The lowest function value on an interval.

Closed Interval Method

Find critical numbers inside the interval, evaluate the original function at the critical numbers and endpoints, then compare the y-values.

What points do you check for absolute extrema on [a,b]?

All critical numbers inside the interval plus the endpoints a and b.

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f must have both an absolute maximum and an absolute minimum on that interval.

What conditions are required for the Extreme Value Theorem?

f must be continuous on the closed interval [a,b].

Does the Extreme Value Theorem tell you where the extrema occur?

No. It only guarantees that they exist.

What does f'(x)>0 mean?

f is increasing.

What does f'(x)

f is decreasing.

First Derivative Test

A method that uses changes in the sign of f' to determine local maxima and minima.

What does f' changing from positive to negative mean?

f has a local maximum.

What does f' changing from negative to positive mean?

f has a local minimum.

What if f' is positive on both sides of a critical number?

There is no local maximum or minimum there.

What if f' is negative on both sides of a critical number?

There is no local maximum or minimum there.

Does f'(c)=0 automatically mean there is a maximum or minimum?

No. The sign of f' must change.

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), then there is at least one c where f'(c)=(f(b)-f(a))/(b-a).

What are the two conditions for the Mean Value Theorem?

f must be continuous on [a,b] and differentiable on (a,b).

What does the Mean Value Theorem guarantee?

A tangent line somewhere inside the interval has the same slope as the secant line connecting the endpoints.

Average rate of change formula

(f(b)-f(a))/(b-a)

What equation do you solve in an MVT problem?

f'(c)=(f(b)-f(a))/(b-a)

Rolle's Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there is at least one c where f'(c)=0.

What extra condition does Rolle's Theorem require?

f(a)=f(b)

What slope does Rolle's Theorem guarantee?

A slope of 0.

How is Rolle's Theorem related to the Mean Value Theorem?

It is a special case of the Mean Value Theorem where the average slope is 0.

What can make the Mean Value Theorem fail?

A discontinuity on [a,b] or a point where f is not differentiable inside (a,b).

Can a cusp prevent MVT from being used?

Yes, because the function is not differentiable at the cusp.

Can a corner prevent MVT from being used?

Yes, because the function is not differentiable at the corner.

Can a vertical tangent prevent MVT from being used?

Yes, because the derivative does not exist there.

Concave Up

The graph bends upward like a cup, and f''(x)>0.

Concave Down

The graph bends downward like a cap, and f''(x)

What does f''(x)>0 mean?

f is concave up.

What does f''(x)

f is concave down.

Inflection Point

A point where the concavity of f changes.

What must happen for an inflection point to exist?

f must change from concave up to concave down or from concave down to concave up.

Does f''(c)=0 automatically mean c is an inflection point?

No. The sign of f'' must change.

What does f' increasing mean about f''?

f''>0.

What does f' decreasing mean about f''?

f''

Second Derivative Test

A test for local extrema using f'' when f'(c)=0.

If f'(c)=0 and f''(c)>0, what happens?

f has a local minimum at c.

If f'(c)=0 and f''(c)

f has a local maximum at c.

If f'(c)=0 and f''(c)=0, what does the Second Derivative Test say?

The test is inconclusive.

What should you use if the Second Derivative Test is inconclusive?

The First Derivative Test.

What does a local maximum of f' often correspond to on f?

A possible inflection point.

What does a local minimum of f' often correspond to on f?

A possible inflection point.

How do you find where f is increasing from the graph of f'?

Find where the graph of f' is above the x-axis.

How do you find where f is decreasing from the graph of f'?

Find where the graph of f' is below the x-axis.

How do you find local maxima of f from the graph of f'?

Find where f' changes from positive to negative.

How do you find local minima of f from the graph of f'?

Find where f' changes from negative to positive.

How do you find where f is concave up from the graph of f'?

Find where f' is increasing.

How do you find where f is concave down from the graph of f'?

Find where f' is decreasing.

How do you find inflection points of f from the graph of f'?

Find where f' changes from increasing to decreasing or from decreasing to increasing.

How do you find concavity from the graph of f''?

f'' above the x-axis means concave up; f'' below the x-axis means concave down.

How do you find inflection points from the graph of f''?

Find where f'' changes sign.

Limits at Infinity

Limits that describe what happens to f(x) as x approaches positive or negative infinity.

For a rational function, what happens if the numerator degree is less than the denominator degree?

The limit is 0.

For a rational function, what happens if numerator and denominator have the same degree?

The limit is the ratio of the leading coefficients.

For a rational function, what happens if the numerator degree is greater than the denominator degree?

The function usually grows without bound or behaves like a polynomial after division.

Horizontal Asymptote

If lim x→∞ f(x)=L or lim x→-∞ f(x)=L, then y=L is a horizontal asymptote.

Can a function have different horizontal asymptotes on the left and right?

Yes.

What is sqrt(x^2)?

|x|

What does |x| equal when x approaches positive infinity?

x

What does |x| equal when x approaches negative infinity?

-x

Why must you be careful with sqrt(x^2) in limits at negative infinity?

Because sqrt(x^2)=|x|, and when x is negative, |x|=-x.

Curve Sketching

Using information about f, f', f'', extrema, concavity, asymptotes, and end behavior to draw a graph.

First step in curve sketching

Plot all known points.

Second step in curve sketching

Mark discontinuities and asymptotes.

Third step in curve sketching

Find or mark critical numbers.

Fourth step in curve sketching

Determine where f is increasing and decreasing.

Fifth step in curve sketching

Determine local maxima and minima.

Sixth step in curve sketching

Determine concavity.

Seventh step in curve sketching

Find inflection points.

Eighth step in curve sketching

Determine end behavior.

Final step in curve sketching

Connect the graph in a way that satisfies all conditions.

Increasing and concave up

f'>0 and f''>0.

Increasing and concave down

f'>0 and f''

Decreasing and concave up

f'

Decreasing and concave down

f'

What happens to slope when a graph is increasing and concave up?

The slope is positive and becoming more positive.

What happens to slope when a graph is increasing and concave down?

The slope is positive but becoming less positive.

What happens to slope when a graph is decreasing and concave up?

The slope is negative but becoming less negative.

What happens to slope when a graph is decreasing and concave down?

The slope is negative and becoming more negative.

If f'(c) does not exist, what could happen on the graph?

There could be a corner, cusp, vertical tangent, or another point of nondifferentiability.

Does f'(c) DNE automatically mean there is a cusp?

No.

Corner

A point where two one-sided slopes exist but are different, such as f(x)=|x| at x=0.

Cusp

A sharp point where the derivative approaches opposite infinities from the two sides, such as f(x)=x^(2/3).

Vertical Tangent

A point where the tangent line is vertical and the derivative is undefined, such as f(x)=x^(1/3) at x=0.

If f has a local maximum at x=c, what should happen on the graph of f'?

f'(c)=0 or may not exist, and f' usually changes from positive to negative.

If f has a local minimum at x=c, what should happen on the graph of f'?

f'(c)=0 or may not exist, and f' usually changes from negative to positive.

If f has an inflection point at x=c, what usually happens to f'?

f' changes from increasing to decreasing or decreasing to increasing.

If f has an inflection point at x=c, what usually happens to f''?

f'' changes sign.

Relationship between f and f'

f' describes the slope and increasing/decreasing behavior of f.

Relationship between f' and f''

f'' describes whether the slope f' is increasing or decreasing.

Master rule: f'>0

f increasing.

Master rule: f'

f decreasing.

Master rule: f' changes + to -

Local maximum of f.

Master rule: f' changes - to +

Local minimum of f.