Concavity

• Concave up: Second derivative is a positive number.

• Concave down: Second derivative is a negative number.

• To find the concavity find the second derivative and set it equal to zero to find critical numbers. Place critical numbers on a number line and test points in between each region (test in second derivative equation). When the values are positive then that interval is concave upward. When the values are negative then that interval is concaved downward.

P.O.I. (Point of

Inflection)

Definition – The point on a graph where the concavity changes

*Cannot be asymptote or undefined point

How to Find:1. Find the second derivative2. Set the second derivative equal to zero, find the critical

values (f ’(x) =0 or where f(x) or f ’(x) is undefined)3. Place on number line and test regions in the second

derivative4. Locate critical numbers where sign changes POI5. Plug P.O.I. into original equation to find y-value

Absolute Minimum Definition

• Lowest point on the graph/equation

• Check endpoints and local minimum heights in table of values to compare/determine

• Careful with endpoints that are not included, they cannot be answers

Absolute Maximum Definition

• Highest point on the graph/equation

• Check endpoints and local maximum heights in table of values to compare/determine

• Careful with endpoints that are not included, they cannot be answers

Increasing/Decreasing

What is increasing and decreasing?

-When the graphs slope is positive/negative

-Steps1. Find the derivative of the function

2. Set the derivative equal to zero, find critical numbers

3. Test regions on the number line into the derivative (slope)

4. If regions are positive its increasing, if negative its decreasing

Local Maximum Definition

• The highest point in a neighborhood of points

• Most cases f’(x)=0, but f’(x) can be undefined there like at a cusp or corner

• Slope goes from positive to negative

Local Minimum Definition

• The lowest point in a neighborhood of points

• Most cases f’(x)=0, but f’(x) can be undefined there like at a cusp or corner

• Slope goes from negative to positive

Finding Relative Extrema

1. Find the derivative

2. Set derivative = 0 and find critical numbers

3. Set up a # line with critical #s on it

4. Test each section by plugging the #s surrounding critical #s into the derivative

5. When there is a change in signs you have relative extrema (negative – positive is minimum) ; (positive – negative is maximum)

6. Plug those critical values into the original to find y value for the relative extrema

Mean Value Theorem

Rolle’sTheorem

• Special case of the Mean Value Theorem

• Case in which, with respect to points where x=a and x=b, f(A) = f(B)

– When the above is true, a point c in between a and b exists so that f ’(C)= 0

• Rolle’s Theorem only exists under the same conditions as the MVT

Second Derivative Test

• Skips doing # line for first derivative to determine if local min/max, can determine using second derivative as well:– Find critical values from the first derivative

– Plug those values into the second derivative

– If the second derivative is +, then concave up, so that value is a local min

– If the second derivative is -, then concave down, so that value is a local max

– If second derivative is 0, then inconclusive

Extreme Value Theorem

• The EVT states that if a function is continuous[a,b], then there must be an absolute max/min (extreme value) on the interval.

Intermediate Value Theorem

• IVT states that if you have a continuousfunction/equation [a,b] then there must exist some c value where f(a)≤f(c)≤f(b).

• Basically states if continuous, all values in between f(a) and f(b) must be reached

• Can help prove you have a root if f(a) is + and f(b) is – or vice versa.

Fermat’s Theorem

• If there is a local min/max at some value x=#, and f ’(#) exists, then f’(#)=0