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Points of Inflection

The max and min on a graph of the derivative are the what of the original function

Points of Inflection and the Second Derivative

If (c,f(c)) is a point of inflection on the graph of f, then either the second derivative at c is zero or it does not exist.

Second derivative test

If the second derivative is greater than zero then the function has a relative minimum

Def. of Concavity

Let f be differentiable on an open interval I. The graph of f is CONCAVE UPWARD on I when f' is increasing on the interval and CONCAVE DOWNWARD on I when f' is decreasing on the interval.

Test of Concavity

Let f be a function whose second deriviative exists on an open interval I.

intermediate value theorem (IVT)

suppose f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b). Then there exists a number "e" such that f(e)=N.

Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that

Extreme Value Theorem (EVT)

If f is continous on the interval x= [a,b] there must be both an absolute min and max

when is f''(x) > 0

when f '(x) is increasing

when is f''(x) < 0

when f '(x) is decreasing

concave up

When f '(x) is increasing, f(x) is

concave down

When f '(x) is decreasing, f(x) is

how do you find critical points

find derivative, set equal to zero

plug into second derivative

how do you determine extrema from critical points

point of inflection

When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a

what creates a relative max in f?

f '(x) changes from positive to negative

what creates a relative min in f?

f '(x) changes from negative to positive

saddle point

when f '(x) = o but doesn't change signs

critical point

a point in the domain of the function at which the derivative is zero or undefined.