If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in[-3,1]$, then the…
If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in[-3,1]$, then the ordered pair $(m, M)$ is equal to
$(-3,19)$
$(3,19)$
$(-19,3)$
$(-19,-3)$
Solution
Given that,
$
f(x)=(x-1)^2+3, \quad x \in[-3,1]
$
On differentiating w.r.t. $x$, we get
$
f^{\prime}(x)=2(x-1)
$
For maxima and minima, put $f^{\prime}(x)=0$
$
\begin{aligned}
\Rightarrow & & 2(x-1) & =0 \\
\Rightarrow & & x & =1
\end{aligned}
$
Now, $f^{\prime \prime}(x)=2$, minima $\forall x \in R$
At $\quad x=1$,
$
f(1)=(1-1)^2+3=3
$
At
$
\begin{aligned}
x & =-3, \\
f(-3) & =(-3-1)^2+3=19
\end{aligned}
$
Here, $m=3$ and $M=19$
Hence, required ordered pair is $(3,19)$