Question

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

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

Answer 1

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

$f.g$ are continuous function

$max\left\{f\right(x):x\in [0,1]\}=max\left\{g\right(x):x\in [0,1]\}$

then graphs of both function will intersect at onepoint $x\in [0,1]$

Let that point be $c$

Then $f\left(c\right)=g\left(c\right)$

According to option A

${\left(f\right(c\left)\right)}^2+3f\left(c\right)={\left(g\right(c\left)\right)}^2+3\left(g\right(c\left)\right)$

$\Rightarrow {\left(f\right(c\left)\right)}^2-{\left(g\right(c\left)\right)}^2+3f\left(c\right)-3\left(g\right(c\left)\right)\Rightarrow \left[f\right(c)-g(c\left)\right]\left[f\right(c)+g(c\left)\right]-3\left[f\right(c)-g(c\left)\right]=0$

according to option D

${\left(f\right(c\left)\right)}^2={\left(g\right(c\left)\right)}^2\left[f\right(c)+g(c\left)\right]=0$