Question

For every pair of continuous function $f, g : [ 0, 1 ] \rightarrow R $
such that max $\{ f (x) : x \in [ 0 , 1] \} = max \{ g (x) : x \in [0, 1] \}$ .
The correct statement(s) is (are)

• A. [ $ f \, (c)]^2 + 3 \, f \, (c) = [ g (c) ]^2+ 3 g \, (c) \, for \, some \,c \in [0, 1]$
• B. [ $ f \, (c)]^2 + f \, (c) = [ g (c) ]^2+ 3 g \, (c) \, for \, some \, c \in [0, 1]$
• C. [ $ f \, (c)]^2 + 3 \, f \, (c) = [ g (c) ]^2+ g \, (c) \, for \, some \, c \in [0, 1]$
• D. [ $ f \, (c)]^2 = [ g (c) ]^2+ 3 g \, (c) \, for \, some \, c \in [0, 1]$

Answer 1

Answer: (A), (D)

Let $f(x)$ and $g$ ( $x)$ achieve their maximum value at $x_{1}$ and $x_{2}$ respectively $h(x)=f(x)-g(x)$
$h \left( x _{1}\right)= f \left( x _{1}\right)- g \left( x _{1}\right) \geq 0 $
$h \left( x _{2}\right)= f \left( x _{2}\right)- g \left( x _{2}\right) \leq 0$
$\Rightarrow h(c)=0$ where $c \in[0,1] \Rightarrow f(c)=g(c)$