Question

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

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

Answer 1

The correct option is A $\left(f\right(c){)}^2+3f(c)=(g\left(c\right){)}^2+3g\left(c\right)$ for some $c\in [0,1]$

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