I have an exercise:
Let $f : \mathbb R^n \rightarrow \mathbb R$ and $T : \mathbb R \rightarrow \mathbb R$ be a strictly monotonically increasing transformation. Show that $x*$ is a maximum of$f$ if and only if $x* $is a maximum of the transformed function $T o f.$
I would be very thankful if someone could show me how to solve this exercise as i am preparing for my calculus exam and don´t even know how to approach this exercise.
(Fist $\Rightarrow$) $x^*\in \mathbb{R}^{n}$ is maximum of $~f$ is the same to write $$ f(x^*)\leq f(x), ~\forall x \in\mathbb{R}^n.$$ Since $T$ is a strictly monotonically increasing transformation, then if $$ a\geq b \Rightarrow T(a)\geq T(b). $$ If you take $a = f(x^*)$ and $b=f(x)\in \mathbb{R}^n$, then you have $$ T(f(x^*)) \geq T(f(x)), \forall x\in \mathbb{R}^n $$ which proves the first part.
(Second $\Leftarrow$) $x^*\in \mathbb{R}^{n}$ is maximum of $~T\circ f$ is the same to write $$ T\circ f(x^*)\geq T\circ f(x), ~\forall x \in\mathbb{R}^n.$$ Since $T$ is a strictly monotonically increasing transformation, then it is invertible and if $$ T(a)\geq T(b) \Rightarrow a\geq b. $$ By choosing $a = f(x^*)$ and $b = f(x)$, we have that $$ f(x^*) \geq f(x), \forall x \in \mathbb{R}^n $$ which proves the second part.