Question 1

Graph the function.   $f ( x , y ) = \sqrt { 1.5 - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 2

Compute the integral.   $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { 7 } ( x - 5 y ) \mathrm { d } x \mathrm {~d} y$

Accepted Answer

A) -1,372 
B)  -2 
C)  -686 
D)  172 
E)  -196 
A) -1,372 
B)  -2 
C)  -686 
D)  172 
E)  -196

Question 3

Locate the maximum of the function.   $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 2 x \right) }$

Accepted Answer

A)  $\left( - 1,0 , e ^ { - 1 } \right)$ 
B)  $( 2,0 , e )$ 
C)  $( 1,0 , e )$ 
D)  $( - 1,0 , e )$ 
E)  $( - 2,0 , e )$ 
A)  $\left( - 1,0 , e ^ { - 1 } \right)$ 
B)  $( 2,0 , e )$ 
C)  $( 1,0 , e )$ 
D)  $( - 1,0 , e )$ 
E)  $( - 2,0 , e )$

Question 4

The US Postal Service (USPS) will accept only packages with length plus girth no more than 102 inches. (See the figure.)
     
What are the dimensions of the largest volume package the USPS will accept 
 
Express your answer as an ordered triple in the form $( x , y , z )$ .

Accepted Answer

The answer of The US Postal Service (USPS) will accept...

Question 5

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model.
  $C ( x , y ) = 6,000 + 200 x ^ { 2 } + 75 y ^ { 2 }$  
where x is the reduction in sulfur emissions, y is the reduction in lead emissions (in pounds of pollutant per day), and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $300 per pound of sulfur and $50 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize the net cost (cost minus subsidy) 
 
Enter your answer as an ordered pair in the form $( x , y )$ . Round your answer to three decimal places if necessary.

Accepted Answer

The answer of The cost of controlling emissions at a...

Question 6

Solve the given problem by using substitution.  
Minimize $S = x y + x z + y z$ subject to $x y z = 5$ with $x > 0$ , $y > 0$ , $z > 0$ .

Accepted Answer

A)  $S _ { \min } = 3 \left( 5 ^ { \frac { 2 } { 3 } } \right)$ 
B)  $S _ { \min } = 3 \left( 6 ^ { \frac { 2 } { 3 } } \right)$ 
C)  $S _ { \min } = 3 \left( 4 ^ { \frac { 2 } { 3 } } \right)$ 
D)  $S _ { \min } = 3 \left( 5 ^ { \frac { 1 } { 3 } } \right)$ 
E)  $S _ { \min } = 3 \left( 6 ^ { \frac { 1 } { 3 } } \right)$ 
A)  $S _ { \min } = 3 \left( 5 ^ { \frac { 2 } { 3 } } \right)$ 
B)  $S _ { \min } = 3 \left( 6 ^ { \frac { 2 } { 3 } } \right)$ 
C)  $S _ { \min } = 3 \left( 4 ^ { \frac { 2 } { 3 } } \right)$ 
D)  $S _ { \min } = 3 \left( 5 ^ { \frac { 1 } { 3 } } \right)$ 
E)  $S _ { \min } = 3 \left( 6 ^ { \frac { 1 } { 3 } } \right)$

Question 7

A productivity model at the Handy Gadget Company is
  $P = 15,000 x ^ { 0.1 } y ^ { 0.9 }$  
where P is the number of gadgets the company turns out per month, x is the number of employees at the company, and y is the monthly operating budget in thousands of dollars. Because the company hires part - time workers, it uses anywhere between 43 and 53 workers each month, and its operating budget varies from $13,000 to $17,000 per month. What is the average of the possible numbers gadgets it can turn out per month  (Round the answer to the nearest 1,000 gadgets.)
 
Please enter your answer as a number without the units.

Accepted Answer

The answer of A productivity model at the Handy Gadget...

Question 8

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 2,3 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 2,3 ) }$ when defined.
​ $f ( x , y ) = x ^ { 2 } y ^ { 3 } - x ^ { 3 } y ^ { 2 } - x y$ ​
Enter your answers, separated by a comma.

Accepted Answer

The answer of Calculate \(\left. \frac { \partial f }...

Question 9

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined.   $f ( x , y ) = e ^ { 2 x y }$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = 2 y e ^ { 2 x }$ , $\frac { \partial f } { \partial y } = 2 x e ^ { 2 y }$ 
B)  $\frac { \partial f } { \partial x } = 2 y e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = 2 x e ^ { 2 x y }$ 
C)  $\frac { \partial f } { \partial x } = e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = e ^ { 2 x y }$ 
D)  $\frac { \partial f } { \partial x } = 2 e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = 2 e ^ { 2 x y }$ 
E)  $\frac { \partial f } { \partial x } = 2 y$ , $\frac { \partial f } { \partial y } = 2 x$ 
A)  $\frac { \partial f } { \partial x } = 2 y e ^ { 2 x }$ , $\frac { \partial f } { \partial y } = 2 x e ^ { 2 y }$ 
B)  $\frac { \partial f } { \partial x } = 2 y e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = 2 x e ^ { 2 x y }$ 
C)  $\frac { \partial f } { \partial x } = e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = e ^ { 2 x y }$ 
D)  $\frac { \partial f } { \partial x } = 2 e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = 2 e ^ { 2 x y }$ 
E)  $\frac { \partial f } { \partial x } = 2 y$ , $\frac { \partial f } { \partial y } = 2 x$

Question 10

Use the given tabular representation of the function f to compute f (5, 10) - f (10, 5).
​ $$\begin{array} { | l | l | l | l | l | l | } 
\hline & & \boldsymbol { x } & & & \
\hline & & \mathbf { 5 } & \mathbf { 1 0 } & \mathbf { 1 5 } & \mathbf { 2 0 } \
\hline \boldsymbol { y } & \mathbf { 5 } & - 1 & 105 & 151 & - 5 \
\hline & \mathbf { 1 0 } & - 1 & 179 & 117 & - 10 \
\hline & \mathbf { 1 5 } & - 11 & 216 & 255 & - 16 \
\hline & \mathbf { 2 0 } & - 12 & 257 & 211 & - 15 \
\hline
\end{array}$$

Accepted Answer

The answer of Use the given tabular representation of the...

Question 11

Locate the critical point of the function.   $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$

Accepted Answer

A)  $( 3,2 , - 17 )$ 
B)  $( - 3,2,17 )$ 
C)  $( 6,4,4 )$ 
D)  $( - 6,4,4 )$ 
E)  $( 2 , - 3,17 )$ 
A)  $( 3,2 , - 17 )$ 
B)  $( - 3,2,17 )$ 
C)  $( 6,4,4 )$ 
D)  $( - 6,4,4 )$ 
E)  $( 2 , - 3,17 )$

Question 12

Your weekly cost (in dollars) to manufacture x cars and y trucks is   $C ( x , y ) = 240,000 + 6,000 x + 4,000 y$  
What is the marginal cost of a truck

Accepted Answer

A) $6,000 
B)  $4,000 
C)  $2,000 
D)  $244,000 
E)  $246,000 
A) $6,000 
B)  $4,000 
C)  $2,000 
D)  $244,000 
E)  $246,000

Question 13

The temperature at the point $( x , y )$ on the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1) is given by   $T ( x , y ) = 4 x ^ { 2 } + 8 y ^ { 2 }$  
Find the hottest and coldest points on the square.

Accepted Answer

A) (2, 2), (0, 0) 
B)  (0, 1), (0, 0) 
C)  (1, 1), (1, 0) 
D)  (1, 0), (0, 0) 
E)  (1, 1), (0, 0) 
A) (2, 2), (0, 0) 
B)  (0, 1), (0, 0) 
C)  (1, 1), (1, 0) 
D)  (1, 0), (0, 0) 
E)  (1, 1), (0, 0)

Question 14

Calculate the values of $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ at $( 4,1,2 )$ .   $f ( x , y , z ) = \frac { 320 z } { 7 + y ^ { 0.6 } }$

Accepted Answer

A)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 0$ 
B)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = - 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 40$ 
C)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = - 40$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 0$ 
D)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 40$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 6$ 
E)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = - 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 320$ 
A)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 0$ 
B)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = - 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 40$ 
C)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = - 40$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 0$ 
D)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 40$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 6$ 
E)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,1,2 ) } = 0$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = - 6$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,1,2 ) } = 320$

Question 15

Find the volume under the graph of $z = 5 - x ^ { 2 }$ over the triangle $0 \leq x \leq 3$ and $0 \leq y \leq 3 - x$ .

Accepted Answer

A) 15.75 
B)  42 
C)  22.5 
D)  7.5 
E)  29.25 
A) 15.75 
B)  42 
C)  22.5 
D)  7.5 
E)  29.25

Question 16

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) }$ when defined.   $f ( x , y ) = 960 - 13 x + 6 y + 7 x y$

Accepted Answer

A)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = - 22$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = - 34$ 
B)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 110$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 136$ 
C)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 28$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 5$ 
D)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 22$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 34$ 
E)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 34$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 22$ 
A)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = - 22$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = - 34$ 
B)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 110$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 136$ 
C)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 28$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 5$ 
D)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 22$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 34$ 
E)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 34$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 22$

Question 17

What point on the surface z is closest to the origin    $z = x ^ { 2 } + y - 5$  
Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.

Accepted Answer

A)  $( 2.5,0 , - 2.5 )$ 
B)  $\left( - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ and $\left( 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $( 0,2.5 , - 2.5 )$ 
D)  $\left( 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
E)  $( 0,2.5 , - 2.5 )$ and $( 0,2.5,2.5 )$ 
A)  $( 2.5,0 , - 2.5 )$ 
B)  $\left( - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ and $\left( 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $( 0,2.5 , - 2.5 )$ 
D)  $\left( 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
E)  $( 0,2.5 , - 2.5 )$ and $( 0,2.5,2.5 )$

Graph the function. $f ( x , y ) = \sqrt { 1.5 - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Compute the integral. $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { 7 } ( x - 5 y ) \mathrm { d } x \mathrm {~d} y$

Locate the maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 2 x \right) }$

The US Postal Service (USPS) will accept only packages with length plus girth no more than 102 inches. (See the figure.) What are the dimensions of the largest volume package the USPS will accept Express your answer as an ordered triple in the form $( x , y , z )$ .

Solve the given problem by using substitution. Minimize $S = x y + x z + y z$ subject to $x y z = 5$ with $x > 0$ , $y > 0$ , $z > 0$ .

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 2,3 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 2,3 ) }$ when defined. $f ( x , y ) = x ^ { 2 } y ^ { 3 } - x ^ { 3 } y ^ { 2 } - x y$ Enter your answers, separated by a comma.

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = e ^ { 2 x y }$

Use the given tabular representation of the function f to compute f (5, 10) - f (10, 5). -1 105 151 -5 -1 179 117 -10 -11 216 255 -16 -12 257 211 -15

Locate the critical point of the function. $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$

Your weekly cost (in dollars) to manufacture x cars and y trucks is $C ( x , y ) = 240,000 + 6,000 x + 4,000 y$ What is the marginal cost of a truck

The temperature at the point $( x , y )$ on the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1) is given by $T ( x , y ) = 4 x ^ { 2 } + 8 y ^ { 2 }$ Find the hottest and coldest points on the square.

Calculate the values of $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ at $( 4,1,2 )$ . $f ( x , y , z ) = \frac { 320 z } { 7 + y ^ { 0.6 } }$

Find the volume under the graph of $z = 5 - x ^ { 2 }$ over the triangle $0 \leq x \leq 3$ and $0 \leq y \leq 3 - x$ .

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) }$ when defined. $f ( x , y ) = 960 - 13 x + 6 y + 7 x y$

What point on the surface z is closest to the origin $z = x ^ { 2 } + y - 5$ Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.

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Introduction to the Derivative

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Exam 15: Functions of Several Variables

Graph the function. f(x,y)=1.5−(x2+y2)f ( x , y ) = \sqrt { 1.5 - \left( x ^ { 2 } + y ^ { 2 } \right) }f(x,y)=1.5−(x2+y2)​

Compute the integral. ∫07∫07(x−5y)dx dy\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { 7 } ( x - 5 y ) \mathrm { d } x \mathrm {~d} y∫07​∫07​(x−5y)dx dy

Locate the maximum of the function. f(x,y)=e−(x2+y2+2x)f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 2 x \right) }f(x,y)=e−(x2+y2+2x)

The US Postal Service (USPS) will accept only packages with length plus girth no more than 102 inches. (See the figure.) What are the dimensions of the largest volume package the USPS will accept Express your answer as an ordered triple in the form (x,y,z)( x , y , z )(x,y,z) .

Solve the given problem by using substitution. Minimize S=xy+xz+yzS = x y + x z + y zS=xy+xz+yz subject to xyz=5x y z = 5xyz=5 with x>0x > 0x>0 , y>0y > 0y>0 , z>0z > 0z>0 .

Calculate ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ , and ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ when defined. f(x,y)=e2xyf ( x , y ) = e ^ { 2 x y }f(x,y)=e2xy

Use the given tabular representation of the function f to compute f (5, 10) - f (10, 5). ​ -1 105 151 -5 -1 179 117 -10 -11 216 255 -16 -12 257 211 -15

Locate the critical point of the function. f(x,y)=4−x2−6x−y2+4yf ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 yf(x,y)=4−x2−6x−y2+4y

Your weekly cost (in dollars) to manufacture x cars and y trucks is C(x,y)=240,000+6,000x+4,000yC ( x , y ) = 240,000 + 6,000 x + 4,000 yC(x,y)=240,000+6,000x+4,000y What is the marginal cost of a truck

The temperature at the point (x,y)( x , y )(x,y) on the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1) is given by T(x,y)=4x2+8y2T ( x , y ) = 4 x ^ { 2 } + 8 y ^ { 2 }T(x,y)=4x2+8y2 Find the hottest and coldest points on the square.

Find the volume under the graph of z=5−x2z = 5 - x ^ { 2 }z=5−x2 over the triangle 0≤x≤30 \leq x \leq 30≤x≤3 and 0≤y≤3−x0 \leq y \leq 3 - x0≤y≤3−x .

What point on the surface z is closest to the origin z=x2+y−5z = x ^ { 2 } + y - 5z=x2+y−5 Hint : Minimize the square of the distance from (x,y,z)( x , y , z )(x,y,z) to the origin.

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Trigonometric Models

Filters

Graph the function. $f ( x , y ) = \sqrt { 1.5 - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Compute the integral. $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { 7 } ( x - 5 y ) \mathrm { d } x \mathrm {~d} y$

Locate the maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 2 x \right) }$

The US Postal Service (USPS) will accept only packages with length plus girth no more than 102 inches. (See the figure.) What are the dimensions of the largest volume package the USPS will accept Express your answer as an ordered triple in the form $( x , y , z )$ .

Solve the given problem by using substitution. Minimize $S = x y + x z + y z$ subject to $x y z = 5$ with $x > 0$ , $y > 0$ , $z > 0$ .

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = e ^ { 2 x y }$

Use the given tabular representation of the function f to compute f (5, 10) - f (10, 5). -1 105 151 -5 -1 179 117 -10 -11 216 255 -16 -12 257 211 -15

Locate the critical point of the function. $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$

Your weekly cost (in dollars) to manufacture x cars and y trucks is $C ( x , y ) = 240,000 + 6,000 x + 4,000 y$ What is the marginal cost of a truck

The temperature at the point $( x , y )$ on the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1) is given by $T ( x , y ) = 4 x ^ { 2 } + 8 y ^ { 2 }$ Find the hottest and coldest points on the square.

Find the volume under the graph of $z = 5 - x ^ { 2 }$ over the triangle $0 \leq x \leq 3$ and $0 \leq y \leq 3 - x$ .

What point on the surface z is closest to the origin $z = x ^ { 2 } + y - 5$ Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.