MATH 131-505 Spring 2015
4.3
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4.3 Derivatives and the Shapes of Curves
The Mean Value Theorem If f is a differentiable function on the interval [a, b], then there exists
a number c between a and b such that
f 0 (c) =
f (b) − f (a)
b−a
or, equivalently,
f (b) − f (a) = f 0 (c)(b − a)
We can see that this theorem is reasonable by interpreting it geometrically.
Increasing/Decreasing Test
• If f 0 (x) > 0 on an interval, then f is increasing on that interval.
• If f 0 (x) < 0 on an interval, then f is decreasing on that interval.
The First Derivative Test Suppose that c is a critical number of a continuous function f .
• If f 0 changes from positive to negative at c, then f has a local maximum at c.
• If f 0 changes from negative to positive at c, then f has a local minimum at c.
• If f 0 does not change sign at c (for example, if f 0 is positive on both sides of c or negative on
both sides), then f has no local maximum or minimum at c.
Recall: A function (or its graph) is called concave upward on an interval I if f 0 is an increasing
function on I. It is called concave downward on I if f 0 is decreasing on I.
Concavity Test
• If f 00 (x) > 0 for all x ∈ I, then the graph of f is concave upward on I.
• If f 00 (x) < 0 for all x ∈ I, then the graph of f is concave downward on I.
The Second Derivative Test Suppose f 00 is continuous near c.
• If f 0 (c) = 0 and f 00 (c) > 0, then f has a local minimum at c.
• If f 0 (c) = 0 and f 00 (c) < 0, then f has a local maximum at c.
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MATH 131-505 Spring 2015
4.3
c
Wen
Liu
Note: The Second Derivative Test is inconclusive when f 00 (c) = 0. In other words, at such a point
there might be a maximum, there might be a minimum, or there might be neither. This test also
fails when f 00 (c) does not exist. In such cases the First Derivative Test must be used.
Examples:
1. Suppose f 00 is continuous on (−∞, ∞).
(a) If f 0 (1) = 0 and f 00 (1) = 1, what can you say about f ?
(b) If f 0 (1) = 0 and f 00 (1) = 0, what can you say about f ?
2. Given f (x) = e6x + e−x . Find the local minimum value and the interval on which f is increasing/decreasing and concave up/concave down.
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MATH 131-505 Spring 2015
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Liu
3. (p. 276) Discuss the curve f (x) = x2/3 (6 − x)1/3 with respect to concavity, points of inflection,
and local maxima and minima. Sketch its graph.
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