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The temperature of a point (x,y) in the plane is given by the expression x^2 + y^2 - 4x + 2y - 12x + 14y + 26. What is the temperature of the coldest point in the plane?

 Mar 25, 2024
 #1
avatar+759 
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First, let's simplify the expression by combining like terms. 

\(x^2 -14x + y^2 + 16y + 26\)

 

Ok! Now all we have to do is to take the derivative with respect to x and y and set that to 0!

 

We have 

\(2x - 14 = 0 \\ 2x = 14 \\ x = 7\)

 

and 

 

\(2y + 16 = 0 \\ 2y = -16 \\ y = -8\)

 

Now, all we have to do is to plug in these values for x and y into the orginial function. 

 

We get \((7)^2 - 14(7) + (-8)^2 + 16(-8) + 26 = -87\)

 

The lowest temperature on the plane is -87 degrees, which is quite cold!

 

Thanks! :)

 May 27, 2024

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