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Find all the points at which the direction of fastest change of the function f(x, y) = x2 + y2 − 4x − 2y is i + j.

User Unitario
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1 Answer

3 votes

Answer:

All the points that lie on the line:


y(x)=x-1

Explanation:

In order to find the maximum rate of change, we need to find the gradient of the function. The gradient of a function of two variables is defined to be:


\\abla f =(\partial f)/(\partial x) i + (\partial f)/(\partial y) j

So:


(\partial f)/(\partial x) = 2x-4=2(x-2)\\\\(\partial f)/(\partial y) = 2y-2=2(y-1)

Hence:


\\abla f(x,y)=2(x-2)i+2(y-1)j

Since we need to find all the points at which the direction of fastest change of the function
f(x,y)=x^(2) +y^(2) -4x-2 is
i+j. Then:


2(x-2)i+2(y-1)=ai+aj\\\\a>0

Therefore:


2(x-2)=2(y-1)\\\\Solving\hspace{3}for\hspace{3}y\\\\y=x-1

So, we can conclude, that all the points where the direction of fastest change of
f(x,y) lie on the line:


y(x)=x-1

User Mitchel Sellers
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