For every pair of continuous functions f, g:[0,1] → R such that max {f(x): x ∈[0,1]}=max {g(x):

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5. Continuity and Differentiation

normal

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

$(A)$ $(f(c))^2+3 f(c)=(g(c))^2+3 g(c)$ 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 $c \in[0,1]$

$(D)$ $(f(c))^2=(g(c))^2$ for some $c \in[0,1]$

$(B,D)$

$(B,C)$

$(A,C)$

$(A,D)$

(IIT-2014)

Solution

Consider

$h(x)=f(x)-g(n) \text { Assume } \quad a < b$

$h(a)=\lambda-g(a)>0 $

$h(b)=f(b)-\lambda<0 $

$\text { else if } a > b h(a) < 0 \text { and } h(b) > 0 .$

By intermediate value theorem $\Rightarrow h(c)=0$ $\quad\quad…….(1)$

$(A)$ $\quad( f ( c ))^2+3 f ( c )=( g ( c ))^2+3 g ( c )$

$( f ( c )- g ( c ))( f ( c )+ g ( c )+3)=0$

So there exist a ' $c$ ' : $f(c)-g(c)$

from (1).

Hence A is correct.

$(D)$ Similarly $( f ( c ))^2=( g ( c ))^2$

$(f(c)-g(c))(f(c)+g(c))=0$

$\Rightarrow \quad( D )$ is correct.

$B$ \& $C$ are wrong as by counter eg

If $f(x)=g(x)=\lambda \neq 0$, then

$B \rightarrow \lambda^2+\lambda=\lambda^2+3 \lambda$ is not possible.

$C \rightarrow \lambda^2+3 \lambda=\lambda^2+\lambda$ is not possible.

Standard 12

Mathematics

Which of the following function can satisfy Rolle's theorem ?

normal

If the functions $f ( x )=\frac{ x ^3}{3}+2 bx +\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to

hard

(JEE MAIN-2023)

In $[0, 1]$ Lagrange's mean value theorem is $ NOT$ applicable to

medium

(IIT-2003)

If the function $f(x) = {x^3} – 6{x^2} + ax + b$ satisfies Rolle’s theorem in the interval $[1,\,3]$ and $f'\left( {{{2\sqrt 3 + 1} \over {\sqrt 3 }}} \right) = 0$, then $a =$ …………..

medium

If $c$ is a point at which Rolle's theorem holds for the function, $f(\mathrm{x})=\log _{\mathrm{e}}\left(\frac{\mathrm{x}^{2}+\alpha}{7 \mathrm{x}}\right)$ in the interval $[3,4],$ where $\alpha \in \mathrm{R},$ then $f^{\prime \prime}(\mathrm{c})$ is equal to