Question

For every pair of continuous function $f,g:[0,1]\to R$ 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

Answer: (A), (D)

$\left(f\right(c){)}^2=(g\left(c\right){)}^2$ for some $c\in [0,1]$

Case I :
When both the function attain maximum at same point.
$f\left(k\right)=g\left(k\right)$
So, $\left(f\right(c){)}^2+3f(c)=(g\left(c\right){)}^2+3g\left(c\right)$ for some $c\in [0,1]$
and $\left(f\right(c){)}^2=(g\left(c\right){)}^2$ for some $c\in [0,1]$ are true when $c=k$

Case II :
When both the function attain the maximum at different points.
$f\left(a\right)$ is maximum.
$g\left(b\right)$ is maximum, $a,b\in [0,1]$
Let a function be defined as,
$h\left(x\right)=f\left(x\right)-g\left(x\right)h\left(a\right)=f\left(a\right)-g\left(a\right)>0h\left(b\right)<0$
So, using intermediate value theorem we can say that,
$h\left(c\right)=0;c\in (a,b)f\left(c\right)=g\left(c\right)$
$\left(f\right(c){)}^2+3f(c)=(g\left(c\right){)}^2+3g\left(c\right)$ for some $c\in [0,1]$
and
$\left(f\right(c){)}^2=(g\left(c\right){)}^2$ for some $c\in [0,1]$ are true.

Now to prove or disprove other options,
Let the two function be $f\left(x\right)=g\left(x\right)=k\ne 0$
So, $\left(f\right(c){)}^2+f(c)\ne (g\left(c\right){)}^2+3g\left(c\right)$
$\left(f\right(c){)}^2+3f(c)\ne (g\left(c\right){)}^2+g\left(c\right)$ for any $c\in [0,1]$