Maximising Property

Question

Let $f:\mathbb{R}^2\mapsto\mathbb{R}$ and $g_1,g_2:\mathbb{R}\mapsto\mathbb{R}$ be continuous functions. Is it true that, for any $a_1,a_2\in\mathbb{R}$, $x$ is a maximiser of $$ f(g_1(x+a_1),g_2(x+a_2)) $$ if and only if it is a maximiser of $$ f(g_1(x),g_2(x)) $$ I couldn't find a counterexample (apart from the ones forcing $g_1$ and $g_2$ not be defined in $x+a_1$ and $x+a_2$), so it seems to be true. Any ideas?

Answer 1

Take the function $f$ defined as $f(x,y)=x+y$, and take the functions $g_1,g_2$ which are both piecewise linear functions that interpolate the following points:

• For $g_1$, the points are: $g_1(-1)=0, g_1(0)=3, g_1(1)=0$. Outside $[-1,1]$, the value of $g_1$ is $0$.
• For $g_2$, the points are: $g_2(-1)=0, g_2(0)=1, g_2(1)=0, g_2(9)=0,g_2(10)=2,g_2(11)=0$. Outside of $[-1,11]$, the value of $g_2$ is $0$.

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Clearly, $f(g_1(x), g_2(x))$ has a maximum at $x=0$, since the expression has a value of $4$ at $x=0$, and is smaller everywhere else.

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However, if we define $F(x)=f(g_1(x-100), g_2(x))$, then

1. $F(0)=f(g_1(100), g_2(0))=g_1(100)+g_2(0)=1$
2. $F(10)=f(g_1(-90), g_2(10))=g_1(-90)+g_2(10)=2$

which means that $F$ does not have a maximum at $x=0$, because $F(10)>F(0)$.