Question

Determine the gradient of the function at the given point. F(x,y)= yx, (2,1)

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Darrell

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To determine the gradient of the function F(x, y) = yx at the point (2, 1), we need to find the partial derivatives with respect to x and y.

Given function: F(x, y) = yx

Partial derivative with respect to x:

\frac{\partial F}{\partial x} = y

Partial derivative with respect to y:

\frac{\partial F}{\partial y} = x

Now, we evaluate the partial derivatives at the point (2, 1):

\frac{\partial F}{\partial x} \Bigg|_{(2,1)} = 1

\frac{\partial F}{\partial y} \Bigg|_{(2,1)} = 2

Therefore, the gradient of the function F at the point (2, 1) is given by the vector:

\nabla F(2, 1) = \Bigg( \frac{\partial F}{\partial x} \Bigg|_{(2,1)}, \frac{\partial F}{\partial y} \Bigg|_{(2,1)} \Bigg) = (1, 2)

\boxed{\nabla F(2, 1) = (1, 2)}

Given function:

Partial derivative with respect to x:

Partial derivative with respect to y:

Now, we evaluate the partial derivatives at the point (2, 1):

Therefore, the gradient of the function F at the point (2, 1) is given by the vector:

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