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Level 607

Concavity & the Second Derivative

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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.