Find the minimum or maximum value of the function if` f(x) = 9x^2+12x+2`

Answer

To find the minimum or maximum value of the function \( f(x) = 9x^2 + 12x + 2 \), we will follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function with respect to \( x \).

\[ f'(x) = \frac{d}{dx}(9x^2 + 12x + 2) \]

Using the power rule, we get:

\[ f'(x) = 18x + 12 \]

### Step 2: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:

\[ 18x + 12 = 0 \]

### Step 3: Solve for \( x \)
Now we solve for \( x \):

\[ 18x = -12 \]
\[ x = -\frac{12}{18} = -\frac{2}{3} \]

### Step 4: Determine if it is a minimum or maximum
Next, we need to determine whether this critical point is a minimum or maximum by calculating the second derivative:

\[ f''(x) = \frac{d}{dx}(18x + 12) = 18 \]

Since \( f''(x) = 18 \) is positive, this indicates that the function has a minimum at \( x = -\frac{2}{3} \).

### Step 5: Find the minimum value
Now we substitute \( x = -\frac{2}{3} \) back into the original function to find the minimum value:

\[ f\left(-\frac{2}{3}\right) = 9\left(-\frac{2}{3}\right)^2 + 12\left(-\frac{2}{3}\right) + 2 \]

Calculating each term:

\[ = 9 \cdot \frac{4}{9} - 8 + 2 \]
\[ = 4 - 8 + 2 \]
\[ = -2 \]

Thus, the minimum value of the function \( f(x) = 9x^2 + 12x + 2 \) is \( -2 \).

### Summary
The minimum value of the function is \( -2 \) at \( x = -\frac{2}{3} \).

---