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 .