Find the maximum and minimum values, if any, of the function
$
f(x) = 9x^2 + 12x + 2
$

Q: Find the maximum and minimum values, if any, of the function $ f(x) = 9x^2 + 12x + 2 $

Explanation: Complete the square: \[ f(x) = 9x^2 + 12x + 2 = 9\left(x^2 + \frac{4}{3}x\right) + 2 = 9\left(x + \frac{2}{3}\right)^2 - 4 + 2 = 9\left(x + \frac{2}{3}\right)^2 - 2 \] \[ \text{Minimum value occurs at } x = -\frac{2}{3} \] \[ f\left(-\frac{2}{3}\right) = -2 \] No maximum value exists since the parabola opens upwards.

Question

Find the maximum and minimum values, if any, of the function  
$
f(x) = 9x^2 + 12x + 2
$

Accepted Answer

Explanation: Complete the square:
$$
f(x) = 9x^2 + 12x + 2 = 9\left(x^2 + \frac{4}{3}x\right) + 2
= 9\left(x + \frac{2}{3}\right)^2 - 4 + 2 = 9\left(x + \frac{2}{3}\right)^2 - 2
$$  
$$
\text{Minimum value occurs at } x = -\frac{2}{3}
$$  
$$
f\left(-\frac{2}{3}\right) = -2
$$  
No maximum value exists since the parabola opens upwards.

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