The least value of the function `f(x) = 4 + 4x + (16)/(x)` is :

Options

A. 4
B. 8
C. 20
D. 24

Answer

To find the least value of the function \( f(x) = 4 + 4x + \frac{16}{x} \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the first derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(4 + 4x + \frac{16}{x}) \]

The derivative of a constant is zero, the derivative of \( 4x \) is \( 4 \), and the derivative of \( \frac{16}{x} \) can be found using the power rule:

\[ \frac{d}{dx}\left(\frac{16}{x}\right) = \frac{d}{dx}(16x^{-1}) = -16x^{-2} = -\frac{16}{x^2} \]

So, we have:

\[ f'(x) = 4 - \frac{16}{x^2} \]

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

\[ 4 - \frac{16}{x^2} = 0 \]

Rearranging gives:

\[ \frac{16}{x^2} = 4 \]

Multiplying both sides by \( x^2 \):

\[ 16 = 4x^2 \]

Dividing both sides by 4:

\[ x^2 = 4 \]

Taking the square root:

\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 3: Determine the nature of the critical points
Next, we need to check whether these critical points correspond to a minimum or maximum by finding the second derivative.

\[ f''(x) = \frac{d}{dx}\left(4 - \frac{16}{x^2}\right) \]

The derivative of \( 4 \) is \( 0 \), and using the power rule again for \( -\frac{16}{x^2} \):

\[ f''(x) = 0 + 32x^{-3} = \frac{32}{x^3} \]

### Step 4: Evaluate the second derivative at the critical points
We will evaluate \( f''(x) \) at \( x = 2 \) and \( x = -2 \).

1. For \( x = 2 \):

\[ f''(2) = \frac{32}{2^3} = \frac{32}{8} = 4 \quad (\text{which is } > 0) \]

This indicates that \( x = 2 \) is a local minimum.

2. For \( x = -2 \):

\[ f''(-2) = \frac{32}{(-2)^3} = \frac{32}{-8} = -4 \quad (\text{which is } < 0) \]

This indicates that \( x = -2 \) is a local maximum.

### Step 5: Find the least value of the function
Since we found that \( x = 2 \) is a local minimum, we will evaluate the function at this point to find the least value:

\[ f(2) = 4 + 4(2) + \frac{16}{2} \]

Calculating this gives:

\[ f(2) = 4 + 8 + 8 = 20 \]

Thus, the least value of the function \( f(x) \) is \( 20 \).

### Final Answer
The least value of the function \( f(x) = 4 + 4x + \frac{16}{x} \) is \( \boxed{20} \).