Find and classify the extreme values of
Subject to the constraint
Consider the function (f(x,y)=xy) subject to the constraint [ g(x,y)=\frac{x^{2}}{8}+\frac{y^{2}}{2}-1=0. ] Using Lagrange multipliers, we solve [ \nabla f=\lambda \nabla g. ] Thus, [ (y,x)=\lambda\left(\frac{x}{4},y\right), ] which ___ __________ _____ ___ __________ __________ ______ __________ _______ _______.
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Consider given by
Find f(2, 0, -1).
Find the directional derivative of the function defined by
at the point (1, 2, -1, -2) in the direction .
Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.
) Show that the function f defined by
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in ? Justify your answer.
Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set where
Give an example of a family fi of subsets of a set X which has finite intersection property. Justify your choice of example.
Does Cantor’s intersection theorem hold for the metric space with the standard metric? Justify your answer.