Find the absolute maximum value and the absolute minimum value of… - Vidyadip

Question

Find the absolute maximum value and the absolute minimum value of the function: $f(x)=(x-1)^{2}+3, x \in[-3,1]$

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Answer

Given that $f(x)=(x-1)^{2}+3, x \in[-3,1]$
$\Rightarrow f'(x) = 2(x - 1)$
Now $, f'(x) = 0$
$\Rightarrow 2(x - 1) = 0$
$\Rightarrow x = 1$
Now, we evaluate the value of $'f '$ at critical point $x = 1,$ and at end points of the interval $[-3, 1]$.
$f(1) = (1 - 1)^2 + 3 = 0 + 3 = 3$
$f(-3) = (-3 - 1)^2 + 3 = 16 + 3 = 19$
Therefore, the absolute maximum value of f on $[-3, 1]$ is $19$ occuring at $x = -3$.
And the absolute minimum value of f on $[-3, 1]$ is $3$ occurring at $x = 1$.

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