Relationships between f, f', f"

A point of inflection occurs when...

f changes concavity & f' changes inc/dec & f" changes signs

A relative max occurs when...

f changes from inc. to dec. & f' changes pos. to neg

If f" is negative,

then f is concave down

If f" is positive,

then f is concave up

If f' is negative,

then f is decreasing

If f" is positive,

then f' is increasing

If f is decreasing,

then f' is negative

If f is increasing,

then f' is positive.

If f' increases..

then f(x) is concave up and f"(x) is positive.

Steps to finding POIs

1st: find the 2nd derivative 2nd: set y" = 0 and solve for x 3rd:place the x values on a # line and plug values between the x values into y" 4th: after identifying possible POIs plug the x values back into the original y to make sure the point exists

Steps to finding relative extrema

1st: find the derivative 2nd: set the derivative equal to zero and solve for x 3rd: place the x-values on a # line and plug values between the x values into y' 4th: where y' changes from + to - is a max and where y' changes from - to + is a min

Steps to finding absolute extrema

1st: find the derivative and set equal to zero 2nd: solve for x 3rd: make a candidate test with the x values and the endpoints 4th: the highest y value is the absolute max and the lowest y value is the absolute min

Point of Inflection

A point on the graph of a function at which the graph changes concavity

Concave Up

Intervals in which the second derivative is positive

Concave Down

Intervals on which the second derivative is negative

Second Derivative Test

Used to determine on what intervals a function is concave up/concave down and the points of inflections.

First Derivative Test

Used to determine where a function's graph has a min/max and is increasing or decreasing.

Derivative

the equation for the slopes of a function (used to find the slope of a tangent line).

If c is a critical number and f"(c) is negative,

then f(c) is a relative maximum