Let `f(x)={(-55x,,,xlt-5),(2x^3-3x^2-120x,,,-5lexlt4),(2x^3-3x^2-36x+10,,,xge4):}`
Then interval in which f(x) is monotonically increasing is

Options

A. `(-5,-4)uu(4,oo)`
B. `(-oo,-4)uu(5,oo)`
C. `(-5,4)uu(5,oo)`
D. `(-5,-4)uu(3,oo)`

Answer

To determine the intervals in which the function \( f(x) \) is monotonically increasing, we will follow these steps:

1. **Identify the Function Segments**: The function \( f(x) \) is defined piecewise:
\[ f(x) = \begin{cases} -55x & \text{if } x < -5 \\ 2x^3 - 3x^2 - 120x & \text{if } -5 \leq x < 4 \\ 2x^3 - 3x^2 - 36x + 10 & \text{if } x \geq 4 \end{cases} \]

2. **Differentiate Each Segment**: We will find the derivative \( f'(x) \) for each segment to determine where the function is increasing (i.e., where \( f'(x) > 0 \)).

- For \( x < -5 \):
\[ f'(x) = -55 \]
This derivative is negative, so \( f(x) \) is decreasing in this interval.

- For \( -5 \leq x < 4 \):
\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 120x) = 6x^2 - 6x - 120 \]
We can factor this:
\[ f'(x) = 6(x^2 - x - 20) = 6(x - 5)(x + 4) \]
To find the critical points, set \( f'(x) = 0 \):
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \\ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]
We need to check the sign of \( f'(x) \) in the intervals \( (-5, -4) \) and \( (-4, 4) \).

- For \( x \geq 4 \):
\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 36x + 10) = 6x^2 - 6x - 36 \]
We can factor this:
\[ f'(x) = 6(x^2 - x - 6) = 6(x - 3)(x + 2) \]
To find the critical points, set \( f'(x) = 0 \):
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \\ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

3. **Analyze the Sign of \( f'(x) \)**:
- For the interval \( -5 < x < -4 \):
Choose \( x = -4.5 \):
\[ f'(-4.5) = 6(-4.5 - 5)(-4.5 + 4) = 6(-9.5)(-0.5) > 0 \quad (\text{increasing}) \]
- For the interval \( -4 < x < 4 \):
Choose \( x = 0 \):
\[ f'(0) = 6(0 - 5)(0 + 4) < 0 \quad (\text{decreasing}) \]
- For the interval \( x \geq 4 \):
Choose \( x = 5 \):
\[ f'(5) = 6(5 - 3)(5 + 2) > 0 \quad (\text{increasing}) \]

4. **Conclusion**: The function \( f(x) \) is monotonically increasing in the intervals:
\[ (-5, -4) \quad \text{and} \quad [4, \infty) \]

Thus, the final answer is:
\[ \text{The intervals in which } f(x) \text{ is monotonically increasing are } (-5, -4) \cup [4, \infty). \]