At any point where $f_x = 0$ and $f_y = 0$ , the function has a local maximum or minimum.

Hints

Consider $f(x,y) = x^2 - y^2$ at the origin.
A saddle satisfies both conditions.

Show the answer

False

Why

False. Both partials vanish at a saddle point too, $x^2 - y^2$ at the origin is a minimum along $x$ and a maximum along $y$ . Classifying a critical point needs the second-derivative test, and where its discriminant is zero even that is inconclusive.

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At any point where fx=0f_x = 0fx​=0 and fy=0f_y = 0fy​=0, the function has a local maximum or minimum.

Practise Optimization: Critical Points and Lagrange Multipliers

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At any point where $f_x = 0$ and $f_y = 0$ , the function has a local maximum or minimum.