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ilyas.h
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Homework Statement
In what directions at the point (2, 0) does the function f(x, y) = xy have rate of change -1?[itex]D_{u}(f)(a,b) = \bigtriangledown f(a,b)\cdot (u_{1}, u_{2}) [/itex][itex]f(x,y) = xy[/itex]
[itex](a,b) = (2,0).[/itex]
The Attempt at a Solution
[itex]\frac{\partial f}{\partial x} = y[/itex]
[itex]\frac{\partial f}{\partial y} = x[/itex]
[itex]\bigtriangledown f(2,0) = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})= (y, x) = (0, 2)[/itex]
plugging in:
[itex]D_{u}(xy)(2,0) = \bigtriangledown f(0,2)\cdot (u_{1},u_{2}) = -1[/itex]
[itex](0,2)\cdot (u_{1},u_{2}) = -1[/itex]
[itex]u_{2} = -0.5[/itex]
[itex]u_{1}[/itex] has infinitely many values.
the last line above is the part I am confused about. Are there infinitely many values for u1? thanks.
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