For what values of `x` the function f(x)=`x^(4)-4x^(3)+4x^(2)+40` is monotonically increasing ?

Options

A. `0 lt x lt 1`
B. `1 lt x lt 2`
C. `2 lt x lt 3`
D. `4 lt x lt 5`

Answer

To determine the values of \( x \) for which the function \( f(x) = x^4 - 4x^3 + 4x^2 + 40 \) is monotonically increasing, we need to follow these steps:

### Step 1: Find the derivative of the function
To find when the function is increasing, we first need to compute its derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(x^4 - 4x^3 + 4x^2 + 40) \]

Using the power rule, we differentiate each term:

\[ f'(x) = 4x^3 - 12x^2 + 8x \]

### Step 2: Set the derivative greater than zero
The function \( f(x) \) is monotonically increasing when \( f'(x) > 0 \).

\[ 4x^3 - 12x^2 + 8x > 0 \]

### Step 3: Factor the derivative
We can factor out the common term \( 4x \):

\[ 4x(x^2 - 3x + 2) > 0 \]

Now, we can factor the quadratic expression:

\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

So, we rewrite the inequality:

\[ 4x(x - 1)(x - 2) > 0 \]

### Step 4: Identify critical points
The critical points occur when \( f'(x) = 0 \):

\[ 4x(x - 1)(x - 2) = 0 \]

This gives us the critical points:

\[ x = 0, \quad x = 1, \quad x = 2 \]

### Step 5: Test intervals
We will test the intervals determined by the critical points: \( (-\infty, 0) \), \( (0, 1) \), \( (1, 2) \), and \( (2, \infty) \).

1. **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \)
\[ f'(-1) = 4(-1)((-1) - 1)((-1) - 2) = 4(-1)(-2)(-3) < 0 \]
(Decreasing)

2. **Interval \( (0, 1) \)**: Choose \( x = 0.5 \)
\[ f'(0.5) = 4(0.5)((0.5) - 1)((0.5) - 2) = 4(0.5)(-0.5)(-1.5) > 0 \]
(Increasing)

3. **Interval \( (1, 2) \)**: Choose \( x = 1.5 \)
\[ f'(1.5) = 4(1.5)((1.5) - 1)((1.5) - 2) = 4(1.5)(0.5)(-0.5) < 0 \]
(Decreasing)

4. **Interval \( (2, \infty) \)**: Choose \( x = 3 \)
\[ f'(3) = 4(3)((3) - 1)((3) - 2) = 4(3)(2)(1) > 0 \]
(Increasing)

### Step 6: Conclusion
The function \( f(x) \) is monotonically increasing in the intervals \( (0, 1) \) and \( (2, \infty) \).

Thus, the values of \( x \) for which the function is monotonically increasing are:

\[ x \in (0, 1) \cup (2, \infty) \]