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Wize University Calculus 1 Textbook > Applications of Differentiation

1st and 2nd Derivative Tests

4 Activities

Wize University Calculus 1 Textbook > Applications of Differentiation

Concavity and Inflection Points

3 Activities

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}.}

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Find all critical points of f(x)f(x)
More 1st and 2nd Derivative Tests Questions:
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Derivative tests
Let f(x)=−x3+x2+xf(x) = -x^3+x^2+x and use the first derivative test to find and classify the extreme values.
Since f is a negative cubic and we’re not given an interval, f has no absolute max or min. Here, f’(x)=−3x2+2x+1f’(x)=-3x^2+2x+1, which exists everywhere. Let’s set it equal to zero and solve for x:
0=−3x2+2x+1=−(3x+1)(x−1)→x=1,−1/3.0 = -3x^2+2x+1 = -(3x+1)(x-1) \rightarrow x=1,-1/3.
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}
Absolute Extrema on Closed Interval
Find the absolute maximum of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Absolute Extrema on Closed Interval
Find the absolute extrema of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Practice: Absolute Extrema on Closed Interval
Q.\textbf{Q.} Find the absolute maximum and minimum values of f(x)=6x43−3x13\displaystyle f(x)=6x^{\frac{4}{3}}-3x^{\frac{1}{3}} on the interval [−1,1][-1,1] and then everywhere, using the second derivative test.
Absolute Extrema on Closed Interval
Find the absolute extrema of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Continuity
Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.
Select all that apply.
Derivative Tests
Which of the following five functions are concave up on their whole domain? Select all that apply.
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}
Relative Extrema
Find and classify all local and absolute extrema of f(x)=sin⁡xcos⁡xf\left(x\right)=\sin x\cos x on the interval [0,  2π]\left[0,\ \ 2\pi\right].
Critical Points and Extrema
Let f(x)f(x) be a smooth function (a function that has infinitely many derivatives that exist everywhere) and consider x=ax = a . Which of the following statements are correct? Check off all correct statements.
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: Minimum Value (Similar to April 2018 Q45)
Find the minimum value of f(x)=14x4+13x3f(x)=\frac{1}{4}x^4+\frac{1}{3}x^3
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
Derivative tests
Let f(x)=−x3+x2+xf(x) = -x^3+x^2+x and use the first derivative test to find and classify the extreme values.
Since f is a negative cubic and we’re not given an interval, f has no absolute max or min. Here, f’(x)=−3x2+2x+1f’(x)=-3x^2+2x+1, which exists everywhere. Let’s set it equal to zero and solve for x:
0=−3x2+2x+1=−(3x+1)(x−1)→x=1,−1/3.0 = -3x^2+2x+1 = -(3x+1)(x-1) \rightarrow x=1,-1/3.
Critical points and extrema
Find all local extreme values of the function f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1
Practice: Absolute Extrema on Closed Interval
Q.\textbf{Q.} Find the absolute maximum and minimum values of f(x)=6x43−3x13\displaystyle f(x)=6x^{\frac{4}{3}}-3x^{\frac{1}{3}} on the interval [−1,1][-1,1] and then everywhere, using the second derivative test.
Absolute Extrema on Closed Interval
Find the absolute extrema of f(x)=3x4−4x3\displaystyle f(x)=3x^{4}-4x^{3} on the interval [−1,2][-1,2] using the second derivative test.
Critical Points and Extrema: Max and Min
Find and classify the critical points of f(x)=xex−exf(x)=xe^{x}-e^{x} as relative maxima or minima.
Derivative Tests
Find the local extreme values of h(x)=x4−2x2h(x)=x^4-2x^2 using the second derivative test.
More Concavity and Inflection Points Questions:
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}.}
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?