(1 point) The function f(x, y)=xy(1−2x−6y) has 4 critical points. List them and select the type of critical point. Points should be entered as ordered pairs and listed in increasing lexicographic order. By that we mean that (x, y) comes before (z, w) if x

The points are (0,0) Minimum

(1/6, 1/18) Saddle point

Step-by-step explanation:

To solve the question we have

f(x,y) = xy(1-2x-6y)

Therefore, f(x,y) = xy-2x²y-6xy²

Finding the partial derivative of the above gives

fₓ = y - 4xy-6y²

= x - 2x²-12xy

Therefore, when the above equations are set to 0 we get

y - 4xy-6y² = 0.......(1)

x - 2x²-12xy  = 0......(2)

Dividing equation (1) and (2) by y and x respectively gives

1 - 4x - 6y = 0

1 - 2x - 12y = 0  

Solving gives

x = 0.167 = 1/6

y = 0.056 = 1/18

When we place x in equation (1) we have

y - 4(1/6)y-6y² = 0

y = 0 or y = 1/18

Similarly, x = 0 or 1/6

Therefore, the four critical points are

(0,0) (1/6,1/18)

When x = 0 and y = 0

f(x,y) = 0 Hence this is a minimum

When x = 1/6 and y = 1/18

f (x,y) = 1/324 Which is a  saddle point.

step-by-step explanation:

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.