2.8.E: Exercises
- Page ID
- 157570
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.8 Exercises
Find all of the critical points of the function shown and identify them as local max, local min, or neither. Find the global max and min on the interval.
Find all of the critical points of the function shown and identify them as local max, local min, or neither. Find the global max and min on the interval.
In problems 3 – 8, find all of the critical points and local maximums and minimums of each function.
| 3. \(f(x) = x^2 + 8x + 7\) | 4. \(f(x) = 2x^2 – 12x + 7\) |
| 5. \(f(x) = x^3 – 6x^2 + 5\) | 6. \(f(x) = (x – 1)^2 (x – 3)\) |
| 7. \(f(x) = \ln ( x^2 – 6x + 11 )\) | 8. \(f(x) = 2x^3 – 96x + 42\) |
In problems 9 – 16, find all critical points and global extremes of each function on the given intervals.
| 9. \(f(x) = x^2 – 6x + 5\) on the entire real number line. |
| 10. \(f(x) = 2 – x^3\) on the entire real number line. |
| 11. \(f(x) = x^3 – 3x + 5\) on the entire real number line. |
| 12. \(f(x) = x - e^x\) on the entire real number line. |
| 13. \(f(x) = x^2 – 6x + 5\) on [ –2, 5]. |
| 14. \(f(x) = 2 – x^3\) on [ –2, 1]. |
| 15. \(f(x) = x^3 – 3x + 5\) on [ –2, 1]. |
| 16. \(f(x) = x-e^x\) on [ 1, 2]. |
Suppose \(f(1) = 5\) and \(f '(1) = 0\). What can we conclude about the point (1,5) if
(a) \(f '(x) < 0\) for \(x < 1\), and \(f '(x) > 0\) for \(x > 1\)?
(b) \(f '(x) < 0\) for \(x < 1\), and \(f '(x) < 0\) for \(x > 1\)?
(c) \(f '(x) > 0\) for \(x < 1\), and \(f '(x) < 0\) for \(x > 1\)?
(d) \(f '(x) > 0\) for \(x < 1\), and \(f '(x) > 0\) for \(x > 1\)?
Define \(A(x)\) to be the area bounded between the \(x\)–axis, the graph of \(f\), and a vertical line at \(x\).
(a) At what value of \(x\) is \(A(x)\) minimum?
(b) At what value of \(x\) is \(A(x)\) maximum?
Define \(S(x)\) to be the slope of the line through the points \((0,0)\) and \(( x, f(x) )\) based on the graph of \(f\) shown.
(a) At what value of \(x\) is \(S(x)\) minimum?
(b) At what value of \(x\) is \(S(x)\) maximum?
The graph of the derivative of a continuous function \(f\).
(a) List the critical numbers of \(f\).
(b) For what values of \(x\) does \(f\) have a local maximum?
(c) For what values of \(x\) does \(f\) have a local minimum?
The graph of the derivative of a continuous function \(g\).
(a) List the critical numbers of \(g\).
(b) For what values of \(x\) does \(g\) have a local maximum?
(c) For what values of \(x\) does \(g\) have a local minimum?
In problems 22 – 24, a function and values of \(x\) so that \(f '(x) = 0\) are given. Use the Second Derivative Test to determine whether each point \((x, f(x))\) is a local maximum, a local minimum or neither
| 22. \(f(x) = 2x^3 – 15x^2 + 6, x = 0, 5 \). |
| 23. \(g(x) = x^3 – 3x^2 – 9x + 7, x = –1, 3\). |
| 24. \(h(x) = x^4 – 8x^2 – 2, x = –2, 0, 2 \). |