V. ANALYSIS OF FUNCTIONS AND THEIR GRAPHS;
APPLICATIONS OF THE DERIVATIVE
函數的分析與圖形及導數的應用
5.1 Increase, Decrease, and Concavity
1. Let f be a function that is continuous on a closed interval [a, b] and differentiable on the open
interval (a, b).
(a) If f ' ( x)  0 for every value of x in (a, b), then f increases on [a, b].
(b) If f ' ( x)  0 for every value of x in (a, b), then f decreases on [a, b].
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(c) If f ' ( x)  0 for every value of x in (a, b), then f is constant on [a, b].
Where f increases on interval means f ( x1 )  f ( x2 ) whenever x1  x2 , f decreases on
interval means f ( x1 )  f ( x2 ) whenever x1  x2 , and f is constant means f ( x1 )  f ( x2 )
for all points x1 and x 2 .
2. Let f be twice differentiable on an open interval I.
(a) If f " ( x)  0 on I, then f is concave up on I.
(b) If f " ( x)  0 on I, then f is concave down on I.
Where f is concave up means f ' is increasing on I, and f is concave down means f ' is
decreasing on I.
3. Inflection points ( x0 , f ( x0 )) of f: If f has a tangent line and changes the direction of its
concavity at ( x0 , f ( x0 )) .
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5.2 Extreme value
1. Relative extreme value:
Relative maximum f ( x0 ) at x 0 : If f ( x0 )  f ( x) for all x in an interval containing x 0 .
Relative minimum f ( x0 ) at x 0 : If f ( x0 )  f ( x) for all x in an interval containing x 0 .
2. Critical points: The points where either f ' ( x)  0 or f is not differentiable.
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Stationary points: The critical points where f ' ( x)  0 .
Theorem: If a function f has any relative extreme value, then they occur at critical points.
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3. Second derivative test for relative extreme value:
Suppose that f is twice differentiable at the point x 0 .
(a) If f ' ( x0 )  0 and f "( x0 )  0 , then f has a relative minimum at x 0 .
(b) If f ' ( x0 )  0 and f " ( x0 )  0 , then f has a relative maximum at x 0 .
(c) If f ' ( x0 )  0 and f "( x0 )  0 , then the test is inconclusive; that is, f may have a
relative maximum, a relative minimum, or neither at x 0 .
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4. Absolute extreme values:
Let f be a function with domain D. Then f (c ) is the
(a) absolute maximum value on D if and only if f ( x)  f (c) for all x in D,
(b) absolute minimum value on D if and only if f ( x)  f (c) for all x in D.
The steps for finding the absolute extreme value are:
Step 1. Find the critical points of f.
Step 2. Evaluate f at all critical points and endpoints.
Step 3. Take the largest and smallest of these values.
5.3 Procedure for Graphing y  f (x) by Hand:
Step 1. Find y ' and y" .
Step2. Find the rise and fall of the curve.
Step 3. Determine the concavity of the curve.
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Step 4. Make a summary and show the curve’s general shape.
Step 5. Plot specific points and sketch the curve.
5.4 Rolle’s Theorem; Mean-Value Theorem
1. Rolle’s theorem:
Let f be differentiable on (a, b) and continuous on [a, b]. If f (a )  f (b)  0 , then there
is at least one point c in (a, b) where f ' (c)  0 .
2. Mean-Value theorem:
Let f be differentiable on (a, b) and continuous on [a, b]. Then there is at least one point c
in (a, b) where f ' (c) 
f (b)  f (a)
.
ba
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