Question

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

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

Answer 1

Answer: (A), (D)

Let $f(x)$ and $g$ ( $x)$ achieve their maximum value at $x_1$ and $x_2$ respectively $h(x)=f(x)-g(x)$
$h\left(x_1\right)=f\left(x_1\right)-g\left(x_1\right)\ge 0$
$h\left(x_2\right)=f\left(x_2\right)-g\left(x_2\right)\le 0$
$\Rightarrow h(c)=0$ where $c\in [0,1]\Rightarrow f(c)=g(c)$