9.7: Maxima/Minima Problems
- Page ID
- 144347
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Let \(f(x, y) = x^2 + y^2\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = xy\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = \sqrt{1 - x^2 - y^2}\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = \sqrt{x^2 + y^2}\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = -x^3 + 4xy - 2y^2 + 1\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = 2xy(x + y) + 1\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = x^2 y\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = x^4+y^2\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Let \(f(x, y) = x^2 + y - e^y\). Identify all critical points on the function. Where possible, use the second partial derivative test to categorize each critical point as a maximum, a minimum, or a saddle point. If the second partial derivative test fails at a critical point, analyze the function to determine the function's behavior at the point; clearly justify your conclusions. Confirm your results by plotting the function using a 3D graphing calculator.
- Find the absolute maximum and minimum values of \(f(x, y) = x^2 + y^2 - 2y + 1\) over the region \(\bigl\{(x, y) \,|\, x^2 + y^2 \le 9\bigr\}\).
- Find the absolute maximum and minimum values of \(f(x, y) = xy - x - 3y\) over the triangular region bounded by the vertices \((0, 0)\), \((0, 4)\), and \((5, 0)\).
- Find the absolute maximum and minimum values of \(f(x, y) = \dfrac{-2y}{x^2 + y^2 + 1}\) over the region \(\bigl\{(x, y) \,|\, x^2 + y^2 \le 4\bigr\}\).
- Find the absolute maximum and minimum values of \(f(x, y) = x^2 y\) over the region in the first quadrant bounded by \(x = 0\), \(y = 0\), \(x^2 + y^2 = 1\), and \(x^2 + y^2 = 4\).
- Miguel needs to construct a six-sided rectangular box with a volume of \(2\) cubic meters. What should the dimensions of the box be to minimize the box's surface area?
- A company manufactures two types of athletic shoes: jogging shoes and cross-trainers. The total revenue from selling \(x\) units of jogging shoes and \(y\) units of cross-trainers is given by \(R(x, y) = -5x^2 - 8y^2 - 2xy + 42x + 102y\), where \(x\) and \(y\) are in thousands of units. Find the values of \(x\) and \(y\) to maximize the company's total revenue.