Find the gradient vector and level curve of the following functions at the given point and show that the gradient is perpendicular to the level curve through the point. a) f(x,y) = x; (2,3) b) f(x,y) = x2+y2; (2,1) c) f(x,y) = x2-y; (2,1) I've already found the gradient vectors and level curves of all of them and also know that to be perpendicular to each-other, their dot product have to be zero, but I cannot find out how to prove that the dot product is zero. Thank you!

Question

Find the gradient vector and level curve of the following functions at the given point and show that the gradient is perpendicular to the level curve through the point.

a) f(x,y) = x; (2,3)

b) f(x,y) = x2+y2; (2,1)

c) f(x,y) = x2-y; (2,1)

I've already found the gradient vectors and level curves of all of them and also know that to be perpendicular to each-other, their dot product have to be zero, but I cannot find out how to prove that the dot product is zero.

Thank you!

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