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

JEE Advanced 2014Paper 1

Exam: JEE Advanced
Year: 2014
Paper: Paper 1
Subject: Mathematics
Chapter: Application of Derivatives

JEE Advanced 2014 · Paper 1 · Mathematics · Application of Derivatives

For every pair of continuous functions f, g: [0, 1] → R such that...

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

then the correct statement (s) is (are)

Options

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

Correct answer

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

Step-by-step solution

Let

f

\left(x\right)

and

\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)

<br >

h\left({x}_{1}\right) = f\left({x}_{1}\right) - g\left({x}_{1}\right) \ge 0

<br >

h\left({x}_{2}\right) = f\left({x}_{2}\right) - g\left({x}_{2}\right) \le 0

<br >

⇒ h\left(c\right) = 0

where

c \in \left[0, 1\right] ⇒ f\left(c\right) = g\left(c\right) .

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