Find the maximum and minimum value, f(x) = (2x - 1)2 + 3 - Vidyadip

Question

Find the maximum and minimum value, f(x) = (2x - 1)² + 3

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Answer

It is given that f(x) = (2x - 1)² + 3
Now, we can see that (2x - 1)² $\ge$ 0 for every x $\in$ R
$\Rightarrow$ f(x) = (2x - 1)² + 3 $\ge$ 3 for every x $\in$ R
The minimum value of f is attained when 2x - 1 = 0
2x - 1 = 0
$\Rightarrow \mathrm{x}=\frac{1}{2}$
Then, Minimum value of $f=f\left(\frac{1}{2}\right)=\left(2 \cdot \frac{1}{2}-1\right)^{2}+3=3$
Now, f' (x) = 4x - 2 = 0, $\Rightarrow$ x = $\frac12$ is the only critical point which is a minimum.
Therefore, function f does not have a maximum value.

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