Question

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

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

Answer 1

Answer: (B), (D)

$\left(f\right(c){)}^2+3f(c)=(g\left(c\right){)}^2+3g\left(c\right)\text{for some}cϵ[0,1]$

Let f and g be maximum at $c_1$ and $c_2$ respectively, $c_1,c_2ϵ(0,1)$.
Let $h\left(x\right)=f\left(x\right)-g\left(x\right)$
Now $h\left(c_1\right)=f\left(c_1\right)-g\left(c_1\right)=+ve$
and $h\left(c_2\right)=f\left(c_2\right)-g\left(c_2\right)=-ve$
$\therefore h\left(x\right)=0$ has at least one root in $c_1,c_2$
$\therefore f\left(x\right)=g\left(x\right)\text{for some}x=cϵ(c_1,c_2)$
$\therefore f\left(c\right)=g\left(c\right)\text{for some}cϵ(0,1)$
clearly (A, D) are correct