Find the maximum and minimum values, if any, of the function
$
f(x) = (2x - 1)^2 + 3
$

Q: Find the maximum and minimum values, if any, of the function $ f(x) = (2x - 1)^2 + 3 $

Explanation: We have: \[ (2x - 1)^2 \geq 0 \text{ for all } x \in \mathbb{R} \] \[ f(x) = (2x - 1)^2 + 3 \geq 3 \text{ for all } x \in \mathbb{R} \] Minimum value occurs when $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$. \[ \text{Minimum value of } f\left(\frac{1}{2}\right) = (2 \cdot \frac{1}{2} - 1)^2 + 3 = 3 \] No maximum value exists since the function is unbounded above.

Question

Find the maximum and minimum values, if any, of the function  
$
f(x) = (2x - 1)^2 + 3
$

Accepted Answer

Explanation: We have:  
$$
(2x - 1)^2 \geq 0 \text{ for all } x \in \mathbb{R}
$$  
$$
f(x) = (2x - 1)^2 + 3 \geq 3 \text{ for all } x \in \mathbb{R}
$$  
Minimum value occurs when $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$.  
$$
\text{Minimum value of } f\left(\frac{1}{2}\right) = (2 \cdot \frac{1}{2} - 1)^2 + 3 = 3
$$  
No maximum value exists since the function is unbounded above.

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