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The concepts of increasing and decreasing functions, the increasing/decreasing test, and the first derivative test. It includes examples, exercises, and instructions on how to determine intervals where functions are increasing or decreasing, and how to find relative and absolute extrema.
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MATH 2250 Section 4.3 - Increasing/Decreasing Functions and First Derivative Test Definitions A function f is an increasing function if the y-values on the graph increase as you go from left to right.
A function f is an decreasing function if the y-values on the graph decrease as you go from left to right.
Example. f (x) = ex^ Example. f (x) = e−x
Most functions switch back and forth from increasing to decreasing.
Example. The graph of f is below. Assume that it includes all of the relevant information about f , and that the domain of f is (−∞, ∞). Determine intervals on which f is increasing and decreasing.
Increasing/Decreasing Test Suppose f is differentiable on an open interval (a, b).
Key observation: If f ′^ changes sign at c, then either f ′(c) = 0 or f ′(c) dne, so c is a critical number of f.
MATH 2250 Section 4.3 - Increasing/Decreasing Functions and First Derivative Test
ln(x) x^3
is increasing and decreasing.
For each of the statements below, decide whether the statement is true for all functions f satisfying all of the conditions described above, for SOME of these functions f , or for NONE of these functions f.
(a) f (x) has a local minimum at x = 2 (b) f ′(3) > 0 (c) f (x) has a local maximum at x = 4 (d) There is exactly one value of a with 3 < a < 7 such that f (x) has a local maximum at x = a.
On the axes below, sketch the graph of y = g′(x). Be sure you pay close attention to each of the following: