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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).
- If f ′(x) > 0 for all x in (a, b), then f is increasing on (a, b)
- If f ′(x) < 0 for all x in (a, b), then f is decreasing on (a, b).
- If f ′(x) = 0 for all x in (a, b), then f is constant on (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.
- Let f (x) be a continuous function whose derivative is f ′(x) = x(x + 3)^2 (x − 5). Deter- mine intervals where f (x) is increasing and decreasing.
- Find intervals where f (x) = 2x^3 + 3x^2 − 12 x + 6 is increasing and decreasing.
MATH 2250 Section 4.3 - Increasing/Decreasing Functions and First Derivative Test
- Find intervals where f (x) =
ln(x) x^3
is increasing and decreasing.
- Determine relative and absolute extrema of f (x) = x^1 /^5 (x − 10).
- Consider functions f satisfying all of the following conditions:
- f (x) is differentiable on the interval 0 < x < 8
- The critical points of f (x) in the interval 0 < x < 8 are x = 2, 4 , and 6. (f (x) has no other critical points in this interval.)
- The table below shows some values of f (x) and of its derivative f ′(x). x 1 3 5 7 f (x) 3 6 11 0 f ′(x) -1?? -
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.
- A portion of the graph of y = g(x) is shown below.
On the axes below, sketch the graph of y = g′(x). Be sure you pay close attention to each of the following:
- where g′^ is defined
- the value of g′(x) near each of x = − 5 , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5
- the sign of g′
- where g′^ is increasing/decreasing/constant
