Math 137 Week 9 Notes – Dr. Paula Smith
4.3
Derivatives of a function give us information about the shape of the graph of the function.
If
f
′
(x) > 0 on an interval, then f(x) is increasing on that interval.
If f
′
(x) < 0 on an interval, then
f(x) is decreasing on that interval.
If f
″
(x) > 0 on an interval, then f(x) is concave upward on
that interval.
If f
″
(x) < 0 on an interval, then f(x) is concave downward on that interval.
We recall that if f
′
(c) = 0, then c is a critical number.
If
f
″
(c) = 0, then (c,f(c)) is an inflection
point.
First derivative test:
If c is a critical number for a continuous function f, then
a) if f
′
changes from positive to negative at c, then f has a local maximum at c
b) if f
′
changes from negative to positive at c, then f has a local minimum at c
c) if f
′
does not change sign across c, then f has no local extremum at c
Second derivative test:
If f
″
is continuous near critical point c, then
a) if f
′
(c) = 0 and f
″
(c) > 0, then f has a local minimum at c
b) if f
′
(c) = 0 and f
″
(c) < 0, then f has a local maximum at c
4.5
From the material in 4.3 it is clearly useful to determine zeros of the function, of its first
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 Spring '08
 SPEZIALE
 Math, Calculus, Derivative, Mathematical analysis, critical number, critical numbers

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