Relative ExtremaFirst Derivative Test - FDTSecond Derivative Test - SDT
Relative Extrema
decreasing
decreasingincreasing
relative maximum
relative maximumrelative
minimumrelative minimum
Critical Points
Critical points are points at which:
•Derivative equals zero (also called stationary point).
•Derivative doesn’t exist.
First Derivative Test
Let f be a differentiable function with f '(c) = 0, then:
•If f '(x) changes from positive to negative, then f has a relative maximum at c.•If f '(x) changes from negative to positive, then f has a relative minimum at c.
•If f '(x) has the same sign from left to right, then f
does not have a relative extremum at c.
Practice Time!!!
Use First Derivative Test to find critical points and state whether they are minimums or maximums.
Critical points
0 2(stationary)
0 2
+ + __
relativemaximum
relativeminimum
Second Derivative Test
Suppose that c is a critical point at which f’(c) = 0, that f(x) exists in a neighborhood of c, and that f(c) exists. Then:
• f has a relative maximum value at c if f”(c) < 0.
•f has a relative minimum value at c if f”(c) > 0.
•If f(c) = 0, the test is not conclusive.
Note: Second derivative test is still used to calculate max and min
Practice Time again !!!Use second derivative test to find extrema
of
critical points = 0, -1, 1
inconclusive
f has a maximum at x = -1
f has a minimum at x = 1
Find extrema of
A.Using first derivative test
B.Using second derivative test
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