Math 216 Calculus 3 Optimization. The first and second ...

Math 216 Calculus 3 Optimization. The first and second derivative tests.

Math 216 Calculus 3 Optimization. The first and second derivative tests.

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Recall: Local maximization and minimization and the derivative

Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) locally maximizes (or locally minimizes) if for all (x, y , z) sufficiently close to (x0, y0, z0), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)).

Math 216 Calculus 3 Optimization. The first and second derivative tests.

2/9

Recall: Local maximization and minimization and the derivative

Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) locally maximizes (or locally minimizes) if for all (x, y , z) sufficiently close to (x0, y0, z0), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)). Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) globally maximizes (or globally minimizes) if for all (x, y , z), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)).

Math 216 Calculus 3 Optimization. The first and second derivative tests.

2/9

Recall: Local maximization and minimization and the derivative

Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) locally maximizes (or locally minimizes) if for all (x, y , z) sufficiently close to (x0, y0, z0), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)). Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) globally maximizes (or globally minimizes) if for all (x, y , z), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)). Just as in the one variable case, derivatives vanish at local extrema.

Theorem

If (x0, y0, z0) locally maximizes f and is in the interior of the domain of f , then [f ](x0, y0, z0) = 0

Math 216 Calculus 3 Optimization. The first and second derivative tests.

2/9

Recall: Local maximization and minimization and the derivative

Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) locally maximizes (or locally minimizes) if for all (x, y , z) sufficiently close to (x0, y0, z0), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)). Let f (x, y , z) be a function of many variables, an input (x0, y0, z0) globally maximizes (or globally minimizes) if for all (x, y , z), f (x, y , z) f (x0, y0, z0) (or f (x, y , z) f (x0, y0, z0)). Just as in the one variable case, derivatives vanish at local extrema.

Theorem

If (x0, y0, z0) locally maximizes f and is in the interior of the domain of f , then [f ](x0, y0, z0) = 0

Application: Let f (x, y ) = x2 + y 2 + 2x - 4y + 4. Find all possibilities to be local maxima / minima of f . Find the global maxima / minima for f on R2, or determine if none exist.

Math 216 Calculus 3 Optimization. The first and second derivative tests.

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