Question

If m and M respectively denote the minimum and maximum value of \[f(x) = {\left( {x - 1} \right)^2} + 3\] for \[x \in \left[ { - 3,1} \right]\] , then the ordered pair \[\left( {m,M} \right)\] is equal to
\[1){\text{ }}\left( { - 3,9} \right)\]
\[2){\text{ }}\left( {3,19} \right)\]
\[3){\text{ }}\left( { - 19,3} \right)\]
\[4){\text{ }}\left( { - 19, - 3} \right)\]

Answer 1

Hint: It is given that \[x \in \left[ { - 3,1} \right]\] so put \[ - 3\] in the first derivative equation and check whether the function is decreasing or increasing. If \[f'(x) > 0\] then \[f\] is increasing. If \[f'(x) < 0\] then \[f\] is decreasing. Then find out the maximum and minimum values of the given function. If the function is decreasing then it would be clear that the given function is maximum at \[x = - 3\] and minimum at \[x = 1\] .Then to find maximum and minimum values of the given function put these values of x in the given. With the help of this you will be able to find the value of the ordered pair.

Complete step-by-step answer:
The given function is \[f(x) = {\left( {x - 1} \right)^2} + 3\] . On differentiating the given function with respect to \['x'\] we get
\[f'(x) = 2{\left( {x - 1} \right)^{2 - 1}}\left( {1 - 0} \right) + 0\]
\[f'(x) = 2\left( {x - 1} \right)\]
It is given that \[x \in \left[ { - 3,1} \right]\] and for \[x = - 3\] ,
\[f'(x) = 2\left( { - 3 - 1} \right)\]
\[f'(x) = 2\left( { - 4} \right)\]
Solving the parenthesis, we get
\[f'(x) = - 8\] which is less than zero \[\left( { < 0} \right)\]
Which implies that \[f(x)\] is decreasing in \[x \in \left[ { - 3,1} \right]\] .
Therefore, the maximum value of \[f(x)\] is at \[x = - 3\] ,
\[f( - 3) = {\left( { - 3 - 1} \right)^2} + 3\]
 \[f( - 3) = {\left( { - 4} \right)^2} + 3\]
Solving the square, we get
\[f( - 3) = 16 + 3\]
\[f( - 3) = 19 = M\]
Where \[M\] denotes the maximum value of the function.
Whereas the minimum value of \[f(x)\] is at \[x = 1\] ,
\[f(1) = {\left( {1 - 1} \right)^2} + 3\]
By solving it further we get
\[f(1) = 3 = m\]
Where the \[m\] denotes the minimum value of the given function.
Therefore , the ordered pair \[\left( {m,M} \right)\] is equal to \[\left( {3,19} \right)\] .
Hence , the correct option is \[2){\text{ }}\left( {3,19} \right)\]
So, the correct answer is “Option 2”.

Note: Keep in mind that if \[f'(x) > 0\] on an open interval , then \[f\] is increasing on the interval . And if \[f'(x) < 0\] on an open interval , then \[f\] is decreasing on the interval . To find out on which interval the function increases or decreases , we take the first derivative of the given function to analyze it to find where it is positive or negative . A function can be represented using ordered pairs .