warm up problems let f (x) = x 3 – 3x + 1. 1) find and classify all critical points. 2) find all...
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Warm up Problems
Let f (x) = x3 – 3x + 1.
1) Find and classify all critical points.
2) Find all inflection points.
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Graph of a Function, Part 2
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Second Derivative Test
If p is a critical point of f (x) and f (p) < 0, then p is a local maximum.
If p is a critical point of f (x) and f (p) > 0, then p is a local minimum.
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Ex. Find and classify all critical points of f (x) = x3 – 5x2 + 3x – 1.
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Def. The absolute maximum (global max) value of a function on an interval is the largest value that the function attains.
Def. The absolute minimum (global min) value of a function on an interval is the smallest value that the function attains.
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Thm. The absolute max. and min. will occur at one of the following:
• the point p where f (p) = 0
• the point p where f (p) is undef.
• an endpoint of the interval
critical points
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Ex. Find the absolute max. and min. values of f (x) = 3x5 – 5x3 – 1 on .4
32,
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Ex. Find the x-coordinate of all local max./min. and absolute max./min. of f (x) = x2 for -2 ≤ x ≤ 0 by graphing.
What about open intervals?
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Ex. Find the x-coordinate of the absolute minimum of f (x) on [0,5]. Justify your answer.
-1
2 4
f '
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Ex. Find the x-coordinate of the absolute maximum of g(x). Justify your answer.
1
0.5
-0.5
-1
-2 -1 1 2
g'
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Ex. For a particle moving along the x-axis, you are given the graph of the velocity below. Assume x(1) = 10.
a) Find the total distance travelled on [1,7].b) Find x(7).c) When is the particle farthest to the left on [1,7]?
2
-2
-4
-6
2 4 6 8
v(t)
(7,-30)
(4,6)