The given temperature function is:

$g(x) = 0.4x^2 - 3.2x + 3.1$

where $x$ represents the time in hours, and the goal is to find the minimum temperature and the time at which it occurs.

To find the minimum temperature, we need to find the vertex of the parabola represented by the quadratic equation. Since the parabola opens upwards (the coefficient of $x^2$ is positive), the vertex will give us the minimum point.

The formula to find the vertex of a parabola $ax^2 + bx + c$ is:

$x_{\text{min}} = \frac{-b}{2a}$

For the given equation:

• $a = 0.4$
• $b = -3.2$

Now, let me calculate the value of $x_{\text{min}}$ and the corresponding temperature $g(x_{\text{min}})$.The minimum temperature occurs at $x = 4$ hours, and the minimum temperature is approximately $-3.3$ degrees Celsius.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do we determine whether a quadratic function has a minimum or maximum?
2. What would happen if the coefficient of $x^2$ were negative?
3. How would we calculate the time if the minimum temperature was known?
4. Can this method be used for any quadratic function to find extreme values?
5. How do changes in the coefficients of the function affect the vertex position?

Tip: Always check the sign of the $x^2$ coefficient to determine whether a quadratic function has a minimum (opens upwards) or maximum (opens downwards).