For every pair of continuous functions f, g :0, 1→R such that max⁡fx:x ∈0, 1=max⁡{gx :x ∈0, 1},then the correct statement (s) is (are)

Q23 / 60 · JEE Advanced 2014 (Paper 1) · Mathematics

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

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

Let

f (x)

and

g (x)

achieve their maximum value at

x_{1}

and

x_{2}

respectively

h (x) = f (x) - g (x)

h (x_{1}) = f (x_{1}) - g (x_{1}) \geq 0

h (x_{2}) = f (x_{2}) - g (x_{2}) \leq 0

\Rightarrow

h (c) = 0

where

c \in [0, 1] \Rightarrow f (c) = g (c) .

Hence, the correct answer is options: 'A', 'D'.