The maximum value of `f(x) = (x -2) (x -3)^(2)` is

Options

A. `4/27`
B. `-4/27`
C. `7/3`
D. 0

Explanation

To find the maximum value of the function \( f(x) = (x - 2)(x - 3)^2 \), we will follow these steps:

### Step 1: Differentiate the Function
We start by differentiating \( f(x) \) using the product rule. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative \( (uv)' = u'v + uv' \).

Let:
- \( u = (x - 2) \)
- \( v = (x - 3)^2 \)

Now, we calculate the derivatives:
- \( u' = 1 \)
- \( v' = 2(x - 3) \)

Using the product rule:
\[ f'(x) = u'v + uv' = 1 \cdot (x - 3)^2 + (x - 2) \cdot 2(x - 3) \]

### Step 2: Simplify the Derivative
Now, we simplify \( f'(x) \):
\[ f'(x) = (x - 3)^2 + 2(x - 2)(x - 3) \]
Expanding \( 2(x - 2)(x - 3) \):
\[ = 2[(x^2 - 5x + 6)] = 2x^2 - 10x + 12 \]
Thus,
\[ f'(x) = (x - 3)^2 + 2x^2 - 10x + 12 \]
Expanding \( (x - 3)^2 \):
\[ = x^2 - 6x + 9 \]
Combining all terms:
\[ f'(x) = x^2 - 6x + 9 + 2x^2 - 10x + 12 = 3x^2 - 16x + 21 \]

### Step 3: Set the Derivative to Zero
To find critical points, we set \( f'(x) = 0 \):
\[ 3x^2 - 16x + 21 = 0 \]

### Step 4: Solve the Quadratic Equation
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3, b = -16, c = 21 \):
\[ x = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 3 \cdot 21}}{2 \cdot 3} \]
Calculating the discriminant:
\[ = 256 - 252 = 4 \]
Thus,
\[ x = \frac{16 \pm 2}{6} \]
Calculating the two values:
\[ x_1 = \frac{18}{6} = 3, \quad x_2 = \frac{14}{6} = \frac{7}{3} \]

### Step 5: Determine Maximum or Minimum
Next, we find the second derivative \( f''(x) \):
\[ f''(x) = 6x - 16 \]
Now we evaluate \( f''(x) \) at the critical points:
1. For \( x = 3 \):
\[ f''(3) = 6(3) - 16 = 18 - 16 = 2 \quad (\text{positive, minimum}) \]
2. For \( x = \frac{7}{3} \):
\[ f''\left(\frac{7}{3}\right) = 6\left(\frac{7}{3}\right) - 16 = 14 - 16 = -2 \quad (\text{negative, maximum}) \]

### Step 6: Calculate the Maximum Value
Now we calculate \( f\left(\frac{7}{3}\right) \):
\[ f\left(\frac{7}{3}\right) = \left(\frac{7}{3} - 2\right)\left(\frac{7}{3} - 3\right)^2 \]
Calculating each term:
\[ = \left(\frac{7}{3} - \frac{6}{3}\right)\left(\frac{7}{3} - \frac{9}{3}\right)^2 = \left(\frac{1}{3}\right)\left(-\frac{2}{3}\right)^2 \]
\[ = \frac{1}{3} \cdot \frac{4}{9} = \frac{4}{27} \]

### Conclusion
Thus, the maximum value of \( f(x) \) is \( \frac{4}{27} \).

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