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Wize University Calculus 1 Textbook > Applications of Differentiation

Concavity and Inflection Points

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Concavity and Inflection Points

While the first derivative tells us about where a function is increasing and decreasing, the second derivative tells us about a functions Concavity. Concavity is the "U" shape of the graph (either up or down).

Concavity

Assume f′(x)f'(x) is differentiable on the interval (a,b)(a, b)
  • If f′′(x)>0f^{\prime\prime}(x)>0 then f(x)f(x) is concave up
  • If f′′(x)<0f^{\prime\prime}(x)<0 then f(x)f(x) is concave down


Inflection Points

An inflection point of a function f(x)f(x) is a point ,in its domain, at which there is a change in concavity in the graph of f(x)f(x)

Wize Tip
An infection point can occur where f(x)′′=0f(x)''=0 or where the second derivative is not defined.

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Example: Concavity and Inflection Points

Determine the intervals of concavity and locate any inflection points for f(x)=ln⁡(1+x2)f\left(x\right)=\ln\left(1+x^2\right)

NOTE: There is an error in the video. From 4:35-4:50, the presenter mistakenly swaps which intervals are concave up and concave down. Otherwise, everything in the video and written solution is correct.
f′(x)=2x1+x2f'\left(x\right)=\dfrac{2x}{1+x^2}


f′′(x)=2(1+x2)−2x(2x)(1+x2)2f''\left(x\right)=\dfrac{2\left(1+x^2\right)-2x\left(2x\right)}{\left(1+x^2\right)^2}


=2−2x2(1+x2)2=\dfrac{2-2x^2}{\left(1+x^2\right)^2}


f′′(x)=0   at   x=±1f''\left(x\right)=0\ \ \ at\ \ \ x=\pm1


f′′(−2)=−625f′′(0)=212f′′(2)=−625<0>0<0Concave DownConcave UpConcave Down\begin{array}{c|c|c} f''(-2)=\dfrac{-6}{25}& f''(0)=\dfrac{2}{1^2}& f''(2)=\dfrac{-6}{25}\\ \\ <0& >0& <0\\ \\ \text{Concave Down}& \text{Concave Up}& \text{Concave Down}\\ \end{array}


Concave down for (−∞,−1)∪(1,∞)(-\infin,-1) \cup(1,\infin)

Concave up for (−1,1)(-1,1)

(−1,ln⁡2) and (1,ln⁡2)(-1,\ln2) \ \text{and} \ (1,\ln2) are inflection points
Find the equation of the tangent line to the function f(x)=ln⁡ xx f(x)=\frac{\ln\ x}{x} at its inflection point.




Extra Practice
Concavity and Inflection Points
Determine the intervals of concavity of the function f(x)=x6−10x4f(x)=x^6-10x^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=0313x=100−1x=20−20\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}
Concavity and inflection points
f′(x)f'(x)plot for a differentiable function is depicted as below. Answer following questions.
Find interval(s) at which ff is concave down.
Concavity and Inflection Points
How many inflection points does the function f(x)=−13x2−5x−e2xf\left(x\right)=-\frac{1}{3x^2}-5x-e^{2x} 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)? (Assume f(x)f\left(x\right) is continuous)
Select all that apply.
Practice: Inflection Point
Practice: Inflection Point
Find 𝑎 and 𝑏 values such that f(x)=ax3+bx2f\left(x\right)=ax^3+bx^2 has an inflection point at (−2, 48)\left(-2,\ 48\right).
Concavity and Inflection Points: Graph of a Derivative
Practice: Graph of a Derivative
Suppose that ff is a differentiable function on the interval [−4,2]\left[-4,2\right]. If the following graph represents the derivative of ff, which of the following statements is correct?
Determine where the following function is concave up/down: f(x)=xexf(x)=xe^x
Q.\textbf{Q.} Determine the interval(s) on which the graph of f(x)=1x2+4\displaystyle f(x)=\frac{1}{x^{2}+4} 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 ff is a differentiable function on the interval (−4,2)(-4,2). If the following graph represents the derivative of ff, which of the following statements is correct?
Concavity and Inflection Points
Let f′′(x)=6x2(x−1)(x2−6)3/2(x−3)3f''(x)=\frac{6x^2\left(x-1\right)}{(x^2-6)^{3/2}(x-3)^3}
Critical Points and Extrema
The following is the graph of f′′(x)f''\left(x\right), 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=0313x=100−1x=20−20\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}
Which function is both always increasing and always concave down on its domain? (Select all that apply)
Practice
f(x)=1x2+1\displaystyle f(x)=\frac{1}{x^2+1}.
The first and the second derivatives are
f′(x)=−2x(x2+1)2 and f′′(x)=6x2−2(x2+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}.}
Practice
f(x)=1x2+1\displaystyle f(x)=\frac{1}{x^2+1}.
The first and the second derivatives are
f′(x)=−2x(x2+1)2 and f′′(x)=6x2−2(x2+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}.}
For a function ff, its 2nd derivative is f′′(x)=8x6−40x5f''\left(x\right)=8x^6-40x^5
Find the values of x for which ff has a point of inflection
Practice: Inflection Point
Practice Question: Inflection Point
Find 𝑎 and 𝑏 values such that f(x)=ax3+bx2f\left(x\right)=ax^3+bx^2 has an inflection point at (−2, 48)\left(-2,\ 48\right).
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)?
i.) f(x)f\left(x\right) has 3 critical points
Concavity and Inflection Points
Find the intervals of increase/decrease and the intervals of concavity for f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Concavity, Inflection Points, Extrema and Intervals
Given f(x)=2−2x+13x3f(x)=2-2x+\frac{1}{3}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)=2x3−2x2+1f(x)=2x^3-2x^2+1
Determine where the following function is concave up/down: f(x)=xexf(x)=xe^x
The graph of the derivative 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) is shown
Where are the local minimums of 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) is shown
Where are the local minimums of f(x)f(x)located? Also list the intervals for which f is concave up or down.
Q.\textbf{Q.} Determine the interval(s) on which the graph of f(x)=1x2+4\displaystyle f(x)=\frac{1}{x^{2}+4} is concave up or down, and the inflection points.
Determine where the following function is concave up/down: f(x)=xexf(x)=xe^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)=x2−2ln⁡xf(x)=x^2-2\ln{x}
Concavity and Inflection Points
Determine the intervals of concavity of the function f(x)=x6−10x4f(x)=x^6-10x^4 and the inflection points of its graph.
If f(x)f(x) is such that f′′(x)=(x2−1)2(x2+1)f''(x)=(x^2-1)^2(x^2+1), then its inflection point(s) occur at
f′(x)f'(x) plot for a differentiable function is depicted below.
Find all the inflection points of ff shown in the plot.
f′(x)f'(x) plot for a differentiable function is depicted below.
a) Find all critical point of ff 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)=x3ln⁡xf(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=e4x2xy=\frac{e^{4x}}{2x}
Which of the following statements is true about the function f(x)=(x2−3)e−xf\left(x\right)=\left(x^2-3\right)e^{-x}?
i.) The function has critical points at x=±3x=\pm\sqrt{3}
ii.) The function has a relative maximum at x=3x=3
Find the inflection points of f(x)=xexf\left(x\right)=xe^x
Concavity and Inflection points
Give the graph of f′(x)f'\left(x\right) below, which of the following statments must be true about the graph of f(x)f\left(x\right)?
(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=x46+3x32−5x22y=\frac{x^4}{6}+\frac{3x^3}{2}-\frac{5x^2}{2}.
Concavity and Inflection Points
How many inflection points does the function f(x)=−13x2−5x−e2xf\left(x\right)=-\frac{1}{3x^2}-5x-e^{2x} have?
Concavity and Inflection Points: Graph of a Derivative
Practice: Graph of a Derivative
Suppose that ff is a differentiable function on the interval [−4,2]\left[-4,2\right]. If the following graph represents the derivative of ff, which of the following statements is correct?
Practice: Inflection Point
Practice: Inflection Point
Find 𝑎 and 𝑏 values such that f(x)=ax3+bx2f\left(x\right)=ax^3+bx^2 has an inflection point at (−2, 48)\left(-2,\ 48\right).
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)? (Assume f(x)f\left(x\right) is continuous)
Select all that apply.
Concavity and Inflection Points
Consider the function f(x)=x4−ax3+2f(x) = x^4 - ax^3 + 2 . For which values of aa does this function have no inflection points, or does such an aa not exist?
Consider the function f(x)=x4−ax3+2f(x) = x^4 - ax^3 + 2 . For which values of aa does this function have no inflection points, or does such an aa not exist?
Concavity and Inflection Points
Which one of the following statements is ALWAYS true for all differentiable functions f(x)f(x) under the conditions given in the statement?
Concavity and inflection points
f′(x)f'(x)plot for a differentiable function is depicted as below. Answer following questions.
Find interval(s) at which ff is concave down.
Concavity and Inflection points
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer following questions.
Find all the inflection points of ff shown in the plot.
Concavity and Inflection Points
Which function is both always increasing and always concave down on its domain?