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Concavity and Inflection Points
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Concavity and Inflection Points
3 Activities
Which one of the following statements is ALWAYS true for all differentiable functions
f
(
x
)
f(x)
f
(
x
)
under the conditions given in the statement?
When
f
′
′
(
a
)
=
0
f''(a) = 0
f
′′
(
a
)
=
0
,
a
a
a
is an inflection point.
Extreme points can only occur at critical points or at endpoints.
When
a
a
a
is a critical point, and the derivative is negative to the left of
a
a
a
, and the derivative is positive to the right of
a
a
a
, then
a
a
a
is a local maximum.
If
f
(
x
)
f(x)
f
(
x
)
is decreasing and concave up, then
f
′
(
a
)
>
0
f'(a) > 0
f
′
(
a
)
>
0
and
f
′
′
(
a
)
>
0
f''(a) > 0
f
′′
(
a
)
>
0
.
The function
x
4
x^4
x
4
has an inflection point at
x
=
0
x = 0
x
=
0
.
I don't know
Check Submission
More Concavity and Inflection Points Questions:
Concavity and Inflection Points
Determine the intervals of concavity of the function
f
(
x
)
=
x
6
−
10
x
4
f(x)=x^6-10x^4
f
(
x
)
=
x
6
−
10
x
4
and the inflection points of its graph.
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f
(
x
)
f
′
(
x
)
f
′
′
(
x
)
x
=
0
3
1
3
x
=
1
0
0
−
1
x
=
2
0
−
2
0
\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
x
=
0
x
=
1
x
=
2
f
(
x
)
3
0
0
f
′
(
x
)
1
0
−
2
f
′′
(
x
)
3
−
1
0
Concavity and inflection points
f
′
(
x
)
f'(x)
f
′
(
x
)
plot for a differentiable function is depicted as below. Answer following questions.
Find interval(s) at which
f
f
f
is concave down.
Concavity and Inflection Points
How many inflection points does the function
f
(
x
)
=
−
1
3
x
2
−
5
x
−
e
2
x
f\left(x\right)=-\frac{1}{3x^2}-5x-e^{2x}
f
(
x
)
=
−
3
x
2
1
−
5
x
−
e
2
x
have?
Max, Min, and Inflection Points
Given the following table of values, which statement(s) must
always
be true about
f
(
x
)
f\left(x\right)
f
(
x
)
? (Assume
f
(
x
)
f\left(x\right)
f
(
x
)
is continuous)
Select all that apply.
Practice: Inflection Point
Practice: Inflection Point
Find 𝑎 and 𝑏 values such that
f
(
x
)
=
a
x
3
+
b
x
2
f\left(x\right)=ax^3+bx^2
f
(
x
)
=
a
x
3
+
b
x
2
has an inflection point at
(
−
2
,
48
)
\left(-2,\ 48\right)
(
−
2
,
48
)
.
Concavity and Inflection Points: Graph of a Derivative
Practice: Graph of a Derivative
Suppose that
f
f
f
is a differentiable function on the interval
[
−
4
,
2
]
\left[-4,2\right]
[
−
4
,
2
]
. If the following graph represents the derivative of
f
f
f
, which of the following statements is correct?
Determine where the following function is concave up/down:
f
(
x
)
=
x
e
x
f(x)=xe^x
f
(
x
)
=
x
e
x
Q.
\textbf{Q.}
Q.
Determine the interval(s) on which the graph of
f
(
x
)
=
1
x
2
+
4
\displaystyle f(x)=\frac{1}{x^{2}+4}
f
(
x
)
=
x
2
+
4
1
is concave up or down, and the inflection points.
Practice: Graph of a Derivative (~F2018 Final Q17) (~F2016 Final Q16)
Practice: Graph of a Derivative
Suppose that
f
f
f
is a differentiable function on the interval
(
−
4
,
2
)
(-4,2)
(
−
4
,
2
)
. If the following graph represents the derivative of
f
f
f
, which of the following statements is correct?
Concavity and Inflection Points
Let
f
′
′
(
x
)
=
6
x
2
(
x
−
1
)
(
x
2
−
6
)
3
/
2
(
x
−
3
)
3
f''(x)=\frac{6x^2\left(x-1\right)}{(x^2-6)^{3/2}(x-3)^3}
f
′′
(
x
)
=
(
x
2
−
6
)
3/2
(
x
−
3
)
3
6
x
2
(
x
−
1
)
Critical Points and Extrema
The following is the graph of
f
′
′
(
x
)
f''\left(x\right)
f
′′
(
x
)
, which of the following statements (if any) is correct?
Critical points and Extrema
Given this table of values, which of the following statements must NOT be true?
f
(
x
)
f
′
(
x
)
f
′
′
(
x
)
x
=
0
3
1
3
x
=
1
0
0
−
1
x
=
2
0
−
2
0
\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}
x
=
0
x
=
1
x
=
2
f
(
x
)
3
0
0
f
′
(
x
)
1
0
−
2
f
′′
(
x
)
3
−
1
0
Which function is both always increasing and always concave down on its domain? (Select all that apply)
Practice
f
(
x
)
=
1
x
2
+
1
\displaystyle f(x)=\frac{1}{x^2+1}
f
(
x
)
=
x
2
+
1
1
.
The first and the second derivatives are
f
′
(
x
)
=
−
2
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
6
x
2
−
2
(
x
2
+
1
)
3
.
\boxed{f^{\prime}(x)=\frac{-2x}{\left(x^{2}+1\right)^{2}}\quad\text{ and }\quad f^{\prime\prime}(x)=\frac{6x^{2}-2}{\left(x^{2}+1\right)^3}.}
f
′
(
x
)
=
(
x
2
+
1
)
2
−
2
x
and
f
′′
(
x
)
=
(
x
2
+
1
)
3
6
x
2
−
2
.
Practice
f
(
x
)
=
1
x
2
+
1
\displaystyle f(x)=\frac{1}{x^2+1}
f
(
x
)
=
x
2
+
1
1
.
The first and the second derivatives are
f
′
(
x
)
=
−
2
x
(
x
2
+
1
)
2
and
f
′
′
(
x
)
=
6
x
2
−
2
(
x
2
+
1
)
3
.
\boxed{f^{\prime}(x)=\frac{-2x}{\left(x^{2}+1\right)^{2}}\quad\text{ and }\quad f^{\prime\prime}(x)=\frac{6x^{2}-2}{\left(x^{2}+1\right)^3}.}
f
′
(
x
)
=
(
x
2
+
1
)
2
−
2
x
and
f
′′
(
x
)
=
(
x
2
+
1
)
3
6
x
2
−
2
.
For a function
f
f
f
, its 2
nd
derivative is
f
′
′
(
x
)
=
8
x
6
−
40
x
5
f''\left(x\right)=8x^6-40x^5
f
′′
(
x
)
=
8
x
6
−
40
x
5
Find the values of x for which
f
f
f
has a point of inflection
Practice: Inflection Point
Practice Question: Inflection Point
Find 𝑎 and 𝑏 values such that
f
(
x
)
=
a
x
3
+
b
x
2
f\left(x\right)=ax^3+bx^2
f
(
x
)
=
a
x
3
+
b
x
2
has an inflection point at
(
−
2
,
48
)
\left(-2,\ 48\right)
(
−
2
,
48
)
.
Practice: Max, Min, and Inflection Points
Practice Question: Max, Min, and Inflection Points
Given the following table of values, which statement(s) must
always
be true about
f
(
x
)
f\left(x\right)
f
(
x
)
?
i.)
f
(
x
)
f\left(x\right)
f
(
x
)
has 3 critical points
Concavity and Inflection Points
Find the intervals of increase/decrease and the intervals of concavity for
f
(
x
)
=
ln
(
x
2
−
1
)
f(x)=\ln(x^2-1)
f
(
x
)
=
ln
(
x
2
−
1
)
Concavity, Inflection Points, Extrema and Intervals
Given
f
(
x
)
=
2
−
2
x
+
1
3
x
3
f(x)=2-2x+\frac{1}{3}x^3
f
(
x
)
=
2
−
2
x
+
3
1
x
3
, find
the intervals of increase and decrease
the local maximum and minimum values
Concavity and Inflection Points
Find all inflection points for the function
f
(
x
)
=
2
x
3
−
2
x
2
+
1
f(x)=2x^3-2x^2+1
f
(
x
)
=
2
x
3
−
2
x
2
+
1
Determine where the following function is concave up/down:
f
(
x
)
=
x
e
x
f(x)=xe^x
f
(
x
)
=
x
e
x
The graph of the derivative
f
′
(
x
)
f^{'}(x)
f
′
(
x
)
is shown
List the intervals for which f is concave up or down.
The graph of the derivative
f
′
(
x
)
f^{'}(x)
f
′
(
x
)
is shown
Where are the local minimums of
f
(
x
)
f(x)
f
(
x
)
located? Also list the intervals for which f is concave up or down.
The graph of the derivative
f
′
(
x
)
f^{'}(x)
f
′
(
x
)
is shown
Where are the local minimums of
f
(
x
)
f(x)
f
(
x
)
located? Also list the intervals for which f is concave up or down.
Q.
\textbf{Q.}
Q.
Determine the interval(s) on which the graph of
f
(
x
)
=
1
x
2
+
4
\displaystyle f(x)=\frac{1}{x^{2}+4}
f
(
x
)
=
x
2
+
4
1
is concave up or down, and the inflection points.
Determine where the following function is concave up/down:
f
(
x
)
=
x
e
x
f(x)=xe^x
f
(
x
)
=
x
e
x
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f
(
x
)
=
x
2
−
2
ln
x
f(x)=x^2-2\ln{x}
f
(
x
)
=
x
2
−
2
ln
x
Concavity and Inflection Points
Determine the intervals of concavity of the function
f
(
x
)
=
x
6
−
10
x
4
f(x)=x^6-10x^4
f
(
x
)
=
x
6
−
10
x
4
and the inflection points of its graph.
If
f
(
x
)
f(x)
f
(
x
)
is such that
f
′
′
(
x
)
=
(
x
2
−
1
)
2
(
x
2
+
1
)
f''(x)=(x^2-1)^2(x^2+1)
f
′′
(
x
)
=
(
x
2
−
1
)
2
(
x
2
+
1
)
, then its inflection point(s) occur at
f
′
(
x
)
f'(x)
f
′
(
x
)
plot for a differentiable function is depicted below.
Find all the inflection points of
f
f
f
shown in the plot.
f
′
(
x
)
f'(x)
f
′
(
x
)
plot for a differentiable function is depicted below.
a) Find all critical point of
f
f
f
in the range shown in the plot.
b) Determine which of the critical points found in the previous part are local max/min.
Intervals of increase and decrease
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f
(
x
)
=
x
3
ln
x
f(x) = x^3 \ln x
f
(
x
)
=
x
3
ln
x
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
y
=
e
4
x
2
x
y=\frac{e^{4x}}{2x}
y
=
2
x
e
4
x
Which of the following statements is true about the function
f
(
x
)
=
(
x
2
−
3
)
e
−
x
f\left(x\right)=\left(x^2-3\right)e^{-x}
f
(
x
)
=
(
x
2
−
3
)
e
−
x
?
i.) The function has critical points at
x
=
±
3
x=\pm\sqrt{3}
x
=
±
3
ii.) The function has a relative maximum at
x
=
3
x=3
x
=
3
Find the inflection points of
f
(
x
)
=
x
e
x
f\left(x\right)=xe^x
f
(
x
)
=
x
e
x
Concavity and Inflection points
Give the graph of
f
′
(
x
)
f'\left(x\right)
f
′
(
x
)
below, which of the following statments must be true about the graph of
f
(
x
)
f\left(x\right)
f
(
x
)
?
(Select all that apply)
Concavity and Infection Points
Find the point(s) of inflection and intervals of concave up and concave down of the graph of
y
=
x
4
6
+
3
x
3
2
−
5
x
2
2
y=\frac{x^4}{6}+\frac{3x^3}{2}-\frac{5x^2}{2}
y
=
6
x
4
+
2
3
x
3
−
2
5
x
2
.
Concavity and Inflection Points
How many inflection points does the function
f
(
x
)
=
−
1
3
x
2
−
5
x
−
e
2
x
f\left(x\right)=-\frac{1}{3x^2}-5x-e^{2x}
f
(
x
)
=
−
3
x
2
1
−
5
x
−
e
2
x
have?
Concavity and Inflection Points: Graph of a Derivative
Practice: Graph of a Derivative
Suppose that
f
f
f
is a differentiable function on the interval
[
−
4
,
2
]
\left[-4,2\right]
[
−
4
,
2
]
. If the following graph represents the derivative of
f
f
f
, which of the following statements is correct?
Practice: Inflection Point
Practice: Inflection Point
Find 𝑎 and 𝑏 values such that
f
(
x
)
=
a
x
3
+
b
x
2
f\left(x\right)=ax^3+bx^2
f
(
x
)
=
a
x
3
+
b
x
2
has an inflection point at
(
−
2
,
48
)
\left(-2,\ 48\right)
(
−
2
,
48
)
.
Max, Min, and Inflection Points
Given the following table of values, which statement(s) must
always
be true about
f
(
x
)
f\left(x\right)
f
(
x
)
? (Assume
f
(
x
)
f\left(x\right)
f
(
x
)
is continuous)
Select all that apply.
Concavity and Inflection Points
Consider the function
f
(
x
)
=
x
4
−
a
x
3
+
2
f(x) = x^4 - ax^3 + 2
f
(
x
)
=
x
4
−
a
x
3
+
2
. For which values of
a
a
a
does this function have no inflection points, or does such an
a
a
a
not exist?
Consider the function
f
(
x
)
=
x
4
−
a
x
3
+
2
f(x) = x^4 - ax^3 + 2
f
(
x
)
=
x
4
−
a
x
3
+
2
. For which values of
a
a
a
does this function have no inflection points, or does such an
a
a
a
not exist?
Concavity and inflection points
f
′
(
x
)
f'(x)
f
′
(
x
)
plot for a differentiable function is depicted as below. Answer following questions.
Find interval(s) at which
f
f
f
is concave down.
Concavity and Inflection points
f
′
(
x
)
f'(x)
f
′
(
x
)
plot for a differentiable function is depicted as below. Answer following questions.
Find all the inflection points of
f
f
f
shown in the plot.
Concavity and Inflection Points
Which function is both always increasing and always concave down on its domain?