The function `f` defined by `f(x)=4x^(4)-2x+1` is increasing for

Options

A. `x lt 1`
B. `x gt0`
C. `x lt 1/2`
D. `x gt 1/2`

Explanation

To determine the intervals where the function \( f(x) = 4x^4 - 2x + 1 \) is increasing, we need to find the derivative of the function and analyze its sign.

### Step 1: Find the derivative of \( f(x) \)

The first step is to differentiate the function \( f(x) \):

\[ f'(x) = \frac{d}{dx}(4x^4 - 2x + 1) \]

Using the power rule, we differentiate each term:

\[ f'(x) = 16x^3 - 2 \]

### Step 2: Set the derivative greater than zero

To find where the function is increasing, we need to set the derivative greater than zero:

\[ f'(x) > 0 \]

This gives us the inequality:

\[ 16x^3 - 2 > 0 \]

### Step 3: Solve the inequality

Rearranging the inequality, we have:

\[ 16x^3 > 2 \]

Dividing both sides by 16:

\[ x^3 > \frac{1}{8} \]

### Step 4: Take the cube root

To solve for \( x \), we take the cube root of both sides:

\[ x > \sqrt[3]{\frac{1}{8}} \]

Since \( \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \), we conclude:

\[ x > \frac{1}{2} \]

### Conclusion

Thus, the function \( f(x) = 4x^4 - 2x + 1 \) is increasing for:

\[ x > \frac{1}{2} \]