Question

Find the absolue maximum value and the absolute minimum value of the following function in the given intervals:

$f\left(x\right)=(x-1{)}^2+3,xϵ[-3,1]$

Answer 1

Given function is $f\left(x\right)=(x-1{)}^2+3,$
$\therefore f^{\prime }\left(x\right)=2(x-1)$
For maxima or minima put $f^{\prime }\left(x\right)=0\Rightarrow 2(x-1)=0\Rightarrow x=1$
Now, we evaluate the value of f at critical point x=1 and at the end points of the interval [-3,1].
$\begin{array}{cc}Atx=1& f\left(1\right)=(1-1{)}^2+3=3\\ ATx=-3& f(-3)=(-3-1{)}^2+3=19\end{array}$
Thus, absolute maximum value is 19 at x=-3 and absolute minimum value is 3 at x=1