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Some questions that I do involving Lagrange Multipliers result in only one critical point. What is a consistent method I can use to show the critical point is a max or a min because it cannot be both. I provided examples to show what I am looking for.

For Example 1: Write the sum of 270 as a sum of 3 numbers so that the sum of products taken 2 at a time is a maximum:

$f(x,y,z) = xy +xz+yz$

$g(x,y,z) = x+y+z = 270$

By LaGrange multipliers, we can easily obtain the point $(90,90,90)$. The question obviously points out that we are looking for a maximum value, but what if we are not told that $(90,90,90)$ is a max? What method do I use to prove it is in fact a maximum?

For Example 2: Let S be the line in the plane through $(−1, 0)$ inclined at 45◦ and let $f(x, y) = x^2 + y^2$ . Find the extrema of f|S and also its extreme values.

$f(x, y) = x^2 + y^2$

$g(x,y) = y-x = 1$ (the equation of the inclined line)

By LaGrange multipliers, we can easily obtain the critical point $(-1/2, 1/2)$ but is it a max or a min?

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If you only have one critical point then you can use the Bordered Hessian technique.

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