Sufficient Condition for Global Optimum

Hi, I have an interesting question posed by my University Mathematics Professor.

It is by definition that the Lagrangian method:
L(x,y) = f(x,y) - λ * [g(x,y) - c]
that f concave and λ * g convex implies that L is concave.
However, is the converse true i.e. L concave implies f concave and λ * g convex? Can we prove it by some other way not by definition such as using PSD NSD principles of their Hessian matrices?

Also, is there a situation whereby L concave does not imply f concave, λ * g convex.

Thank you! :)

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$