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

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

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\[c \in \left[ {0,{\text{ }}1} \right]\].

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

D.\[{\left[ {f\left( c \right)} \right]^2}{\text{ }} = {\text{ }}{\left[ {g\left( c \right)} \right]^2}\]for some\[c \in \left[ {0,{\text{ }}1} \right]\].

Answer
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Hint:For any function \[f\left( x \right)\] if at some point p function is positive and any other point q function is negative that is \[f\left( p \right) > 0\] and \[f\left( q \right) < 0\] then there is at least one root of \[f\left( x \right)\] between p and q.

Complete step-by-step answer:
We are given that for every pair of continuous function \[f,g:\left[ {0,{\text{ }}1} \right] \to R\] such that\[max\{ f\left( x \right):x \in \left[ {0,1} \right]\} = max\{ g\left( x \right):x \in \left[ {0,1} \right]\} \].

Let us take one function at a time. First we take \[f\left( x \right)\].
Let the value of which the function \[f\left( x \right)\] is maximum by ${c_1}$. This is shown as –
\[max\{ f\left( x \right):x \in \left[ {0,1} \right]\} \]=\[f\left( {{c_1}} \right)\]

Similarly, we take the second function \[g\left( x \right)\] and we get,
Let the value of which the function \[g\left( x \right)\] is maximum by ${c_2}$. This is shown as –
\[max\{ g\left( x \right):x \in \left[ {0,1} \right]\} = g({c_2})\]

Now let us take one other function \[h\left( x \right)\] as shown –
\[h\left( x \right)\]=\[f\left( x \right)\]-\[g\left( x \right)\]
Let us find Function \[h\left( x \right)\] for $x = {c_1}$. We get,
\[h\left( {{c_1}} \right)\]=\[f\left( {{c_1}} \right)\]-\[g\left( {{c_1}} \right)\]
Since, \[f\left( {{c_1}} \right)\] is greater than \[g\left( {{c_1}} \right)\] because at $x = {c_1}$ we get maximum value of \[f\left( x \right)\] whereas in case of \[g\left( x \right)\] at $x = {c_1}$ we do not get the maximum value. Therefore,
\[h\left( {{c_1}} \right) > 0\]

Let us find Function \[h\left( x \right)\] for $x = {c_2}$. We get,
\[h\left( {{c_2}} \right)\]=\[f\left( {{c_2}} \right)\]-\[g\left( {{c_2}} \right)\]
Since, \[g\left( {{c_2}} \right)\] is greater than \[f\left( {{c_2}} \right)\] because at $x = {c_2}$ we get maximum value of \[g\left( x \right)\] whereas in case of \[f\left( x \right)\] at \[x = {c_2}\] we do not get the maximum value. Therefore,
\[h\left( {{c_2}} \right) < 0\]
Therefore, for any arbitrary \[c\] which lie in between ${c_1}$ and ${c_2}$ \[h\left( x \right)\] has a root that is –
\[h\left( c \right) = 0\] For \[c \in \left[ {0,{\text{ }}1} \right]\].
Which implies,
\[f(c) - g\left( c \right) = 0\] For \[c \in \left[ {0,{\text{ }}1} \right]\]
\[ \Rightarrow f(c) = g\left( c \right)\]…………. (1)
Now, squaring both sides we get –
\[ \Rightarrow {[f(c)]^2} = {[g\left( c \right)]^2}\]…….. (2)
Multiplying equation (1) by 3 we get,
\[ \Rightarrow 3f(c) = 3g\left( c \right)\]………… (3)
Adding both the equations (2) and (3) we get,
\[{\left[ {f\left( c \right)} \right]^2} + {\text{3}}f\left( c \right) = {\left[ {g\left( c \right)} \right]^2} + 3g\left( c \right)\]…….. (4)
Therefore, from equations (2) and (4) we get that options (A) and (D) are correct options.

So, the correct answer is “Option A and D”.

Note:When we use the property that for any function \[f\left( x \right)\] if at some point p function is positive and any other point q function is negative that is \[f\left( p \right) > 0\] and \[f\left( q \right) < 0\] then there is at least one root of \[f\left( x \right)\] between p and q. It should be noted that the number of roots lying between p and q are atleast one and not equal to one.