Q. If m and M respectively denote the minimum and maximum of $f(x) = (x - 1)^{2} + 3\ for\ x \in \lbrack - 3,1\rbrack,$ then the ordered pair (m, M) is equal to
A) ($- 3$,19)
B) (3,19)
C) $( - 19,3)$
D) $( - 19, - 3)$
Explanation
(b)
Given,
$f(x) = (x - 1)^{2} + 3,\ \ \ x \in \lbrack - 3,\ 1\rbrack$
$\Rightarrow \ \ \ f^{'}(x) = 2(x - 1)$
For maxima and minima, put $f^{'}(x) = 0\ \Rightarrow \ \ x = 1$
Now, $f^{''}(x) = 2$, minima $\forall\ x \in R$
At $x = 1,\ \ \ f(1) = (1 - 1)^{2} + 3 = 3$
At $x\text{=} - 3,\ \ \ \ \ \ \ f( - 3) = ( - 3 - 1)^{2} + 3 = 19$
Here, $m = 3$ and $M = 19$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.