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Second Derivative Test and Concavity of Functions, Exams of Physics

Information on the Second Derivative Test for determining the concavity and inflection points of functions. It includes examples of finding points of inflection and intervals of concavity up and down for various functions.

What you will learn

  • How do you find the points of inflection of a function using the Second Derivative Test?
  • What is the relationship between the first and second derivatives of a function and its concavity?
  • What are the conditions for a function to be concave up or concave down according to the Second Derivative Test?

Typology: Exams

2021/2022

Uploaded on 09/12/2022

gwen
gwen 🇺🇸

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1
f(x) is
concave up
whenever f'(x) is increasing.
f(x) is
concave down
whenever f'(x) is decreasing.
3.4--The 2nd Derivative Test
The test for concavity:
If f"(x) is
positive
for all x in (a,b)
then f(x) is
concave up
in (a,b).
If f"(x) is
negative
for all x in (a,b)
then f(x) is
concave down
in (a,b).
A
point of inflection
occurs where the
concavity changes
.
If (c,
f(c)) is a
point of inflection
, then both #1 and #2 are true:
1) f"(c) is either
zero
or
undefined
.
2) f"(x) changes signs at x
=
c.
If f"(c)
=
0, it
doesn't
guarantee
that f(x) has a POI at x
=
c.
pf3
pf4
pf5
pf8
pf9
pfa

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f(x) is concave up whenever f'(x) is increasing. f(x) is concave down whenever f'(x) is decreasing. The test for concavity: If f"(x) is positive for all x in (a,b) then f(x) is concave up in (a,b). If f"(x) is negative for all x in (a,b) then f(x) is concave down in (a,b). A point of inflection occurs where the concavity changes. If (c, f(c)) is a point of inflection, then both #1 and #2 are true:

  1. f"(c) is either zero or undefined.
  2. f"(x) changes signs at x = c. If f"(c) (^) = 0, it doesn't guarantee that f(x) has a POI at x (^) = c.
  1. f(x) = x 4
  • 4x 3 Find the points of inflection, then determine the intervals where the function is concave up and concave down:
  1. y (^) = ½x 4
  • 2x 3
  • 72x 2
  • 5
  1. A function f is continuous on the closed interval [-1,3] and its derivatives have the values indicated in the table below. (a) Find the x-coordinates of all local extrema of f on (-1,3) and indicate which are maxima and which are minima. Explain. (b) Find all the intervals on which f is concave up or concave down. (c) Find the x-coordinates of any points of inflection. Explain. (d) Sketch the graph of y = f(x) with all the given features.

The 2nd Derivative Test In order to use this test, f'(c) must equal 0. If f"(c) > 0, then f(c) is a MINIMUM. If f"(c) (^) < 0, then f(c) is a MAXIMUM. If f"(c) = 0, then the TEST FAILS. (You'll need to go back and use the 1st Derivative Test.) Find all relative extrema. Use the 2nd Derivative Test whenever possible:

  1. f(x) = (-1/3) x 3
  • 2x 2
  • 5
  1. A function is defined by f(x) = x 3
  • ax 2
  • bx + c, where a, b, and c are constants. f(-1) = 4 is a local maximum, and the graph has a point of inflection at x = 2. Find the value of a, b, and c:
  1. If g(-1) = 3, g'(-1) (^) = 0, and g"(-1) (^) = 5, what can you conclude about the value 3?
  1. Let h be a differentiable function with only one critical number: x = -2. If h(-2) = -4, h'(-3) = -7, and h'(-1) = 0.2, what can you conclude about the value -4?
  2. Based on the graph, find the sign (+,-,0) of each value: f(x) f(-2) _____ f'(-2) _____ f"(-2) _____ f(5) _____ f'(5) _____ f"(5) _____ f(0) _____ f'(0) _____ f"(0) _____
  1. Sketch a continuous curve with the following properties: f(-4) = 1, f(0) = 3, f(4) = 5, f'(-4) = f'(4)^ = 0, f'(x) < 0 for lxl > 4, f'(x) > 0 for lxl < 4, f"(x) > 0 for x < 0, f"(x) < 0 for x > 0.