4.7 Maxima/minima problems  (Page 7/10)

$f(x,y)=1+x^2+y^2$

$f(x,y)={(3x-2)}^2+{(y-4)}^2$

$f(x,y)=x^4+y^4-16xy$

$f(x,y)=15x^3-3xy+15y^3$

$f(x,y)=\sqrt{x^2+y^2+1}$

$f(x,y)=\text{−}{x}^2-5y^2+8x-10y-13$

$f(x,y)=x^2+y^2+2x-6y+6$

$f(x,y)=\sqrt{x^2+y^2}+1$

$f(x,y)=\text{−}{x}^3+4xy-2y^2+1$

$f(x,y)=x^2{y}^2$

$f(x,y)=x^2-6x+y^2+4y-8$

$f(x,y)=2xy+3x+4y$

$f(x,y)=8xy(x+y)+7$

$f(x,y)=x^2+4xy+y^2$

$f(x,y)=x^3+y^3-300x-75y-3$

$f(x,y)=9-x^4{y}^4$

$f(x,y)=7x^{2}y+9x{y}^2$

$f(x,y)=3x^2-2xy+y^2-8y$

$f(x,y)=3x^2+2xy+y^2$

$f(x,y)=y^2+xy+3y+2x+3$

$f(x,y)=x^2+xy+y^2-3x$

$f(x,y)=x^2+2y^2-x^{2}y$

$f(x,y)=x^2+y-e^y$

$f(x,y)=e^{\text{−}(x^2+y^2+2x)}$

$f(x,y)=x^2+xy+y^2-x-y+1$

$f(x,y)=x^2+10xy+y^2$

$f(x,y)=\text{−}{x}^2-5y^2+10x-30y-62$

$f(x,y)=120x+120y-xy-x^2-y^2$

$f(x,y)=2x^2+2xy+y^2+2x-3$

$f(x,y)=x^2+x-3xy+y^3-5$

$f(x,y)=2xy{e}^{\text{−}{x}^2-y^2}$

[T] $f(x,y)=y{e}^x-e^y$

[T] $f(x,y)=x\text{sin}(y)$

[T] $f(x,y)=\text{sin}(x)\text{sin}(y),x\in \left(0,2\pi \right),y\in \left(0,2\pi \right)$

$f(x,y)=xy-x-3y;$ $R$ is the triangular region with vertices $\left(0,0\right),\left(0,4\right),\text{and}\left(5,0\right).$

Find the absolute maximum and minimum values of $f(x,y)=x^2+y^2-2y+1$ on the region $R=\left\{(x,y)|x^2+y^2\le 4\right\}.$

$f(x,y)=x^3-3xy-y^3$ on $R=\{(x,y)\text{:}\mathrm{-2}\le x\le 2,\mathrm{-2}\le y\le 2\}$

$f(x,y)=\frac{\mathrm{-2}y}{x^2+y^2+1}$ on $R=\left\{(x,y)\text{:}{x}^2+y^2\le 4\right\}$

Find three positive numbers the sum of which is $27,$ such that the sum of their squares is as small as possible.

Find the points on the surface $x^2-yz=5$ that are closest to the origin.

Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the line $x+y+z=1.$

The sum of the length and the girth (perimeter of a cross-section) of a package carried by a delivery service cannot exceed $108$ in. Find the dimensions of the rectangular package of largest volume that can be sent.

A cardboard box without a lid is to be made with a volume of $4$ ft ³ . Find the dimensions of the box that requires the least amount of cardboard.

Find the point on the surface $f(x,y)=x^2+y^2+10$ nearest the plane $x+2y-z=0.$ Identify the point on the plane.

Find the point in the plane $2x-y+2z=16$ that is closest to the origin.

A company manufactures two types of athletic shoes: jogging shoes and cross-trainers. The total revenue from $x$ units of jogging shoes and $y$ units of cross-trainers is given by $R(x,y)=\mathrm{-5}{x}^2-8y^2-2xy+42x+102y,$ where $x$ and $y$ are in thousands of units. Find the values of x and y to maximize the total revenue.

A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed $96$ in. Find the dimensions of the box that meets this condition and has the largest volume.

Find the maximum volume of a cylindrical soda can such that the sum of its height and circumference is $120$ cm.