Quiz about Local Maxima and Local Minima

Question 1

A differentiable function z=f(x,y) can have a maximum or minimum at an interior point (x0,y0) of S only if its is a stationary point, that is if the point x,y=(x0,y0) satisfies

Accepted Answer

$$f_x=0,\ f_y=0$$

Answer

$$f_x=0$$

Answer

$$f_y=0$$

Answer

$$f_{xx}=0,\ f_{yy}=0$$

Question 2

If f $$f_{xx}f_{yy}-\left(f_{xy}\right)^2>0$$ and fxx<0 , then the point(x0,y0) is a

Accepted Answer

Local maximum point

Answer

Local minimum point

Answer

Saddle point

Answer

No conclusion

Question 3

If $$f_{xx}f_{yy}-\left(f_{xy}\right)^2>0$$ and fxx>0 , then the point(x0,y0) is a

Accepted Answer

Local minimum point

Answer

Local maximum point

Answer

Saddle point

Answer

No conclusion

Question 4

If $$f_{xx}f_{yy}-\left(f_{xy}\right)^2<0$$ , then the point(x0,y0) is a

Accepted Answer

Saddle point

Answer

Local minimum point

Answer

Local maximum point

Answer

No conclusion

Question 5

Find the critical point of the function $$f(x,y)=2x^2+y^2-2xy+5x-3y+1$$ . Which of the following points is a critical point (x,y)?

Accepted Answer

(-1, 1/2)

Answer

(1, 1)

Answer

(0, 0)

Answer

(-1, -1)

Question 6

Find fx for $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

$$3x^2-y$$

Answer

$$3y^2-x$$

Answer

$$3x^2+3y^2$$

Answer

$$3x^2+3y^2-x-y$$

Question 7

Find fy for $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

$$3y^2-x$$

Answer

$$3x^2-y$$

Answer

$$3x^2+3y^2$$

Answer

$$3x^2+3y^2-x-y$$

Question 8

Find critical point for $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

(0,0), (1/3, 1/3)

Answer

(0,1) , (1,3)

Answer

(0,0), (1,1)

Answer

(0,0), (3,3)

Question 9

Find fxx for $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

6x

Answer

6y

Answer

$$3x^2+3y^2$$

Answer

$$3x^2+3y^2-x-y$$

Question 10

Find fyy for $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

6y

Answer

6x

Answer

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

Answer

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

Question 11

Find fxy for $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

-1

Answer

6x

Answer

6y

Answer

$$3x^2+3y^2$$

Question 12

Find $$f_{xx}f_{yy}-\left(f_{xy}\right)^2$$ for equation $$f(x,y)=x^3+y^3-xy$$

Accepted Answer

36y-1

Answer

6y

Answer

36xy-1

Answer

-1

Question 13

The equation $$f(x,y)=x^3+y^3-xy$$ has two stationary point (0,0) and (1/3,1/3). For point (0, 0) the point is

Accepted Answer

Saddle point

Answer

Minimum point

Answer

Maximum point

Answer

No conclusion

Question 14

The equation $$f(x,y)=x^3+y^3-xy$$ has two stationary point (0,0) and (1/3,1/3). For point (1/3, 1/3) the point is

Accepted Answer

Minimum point

Answer

Saddle point

Answer

Maximum point

Answer

No conclusion

Question 15

Based on this figure, point A is ____ and point B is _____

Accepted Answer

Local maximum, local minimum

Answer

Local minimum, local maximum

Answer

Local maximum, saddle point

Answer

Saddle point, local minimum

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