`f (x)= x^(6) -x-1, x in [1,2].` Consider the following statements :

Options

A. f is increasing on `[1,2]`
B. f has a root in `[1,2]`
C. f is decreasing on `[1,2]`
D. f has no root in `[1,2]`

Answer

To solve the problem, we need to analyze the function \( f(x) = x^6 - x - 1 \) on the interval \([1, 2]\) and determine the validity of the given statements regarding the function's behavior.

### Step 1: Find the derivative of \( f(x) \)
We start by differentiating \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^6 - x - 1) = 6x^5 - 1 \]

**Hint:** To find whether the function is increasing or decreasing, we need to analyze the sign of the derivative.

### Step 2: Analyze the derivative on the interval \([1, 2]\)
Next, we evaluate \( f'(x) \) at the endpoints of the interval:
- For \( x = 1 \):
\[ f'(1) = 6(1)^5 - 1 = 6 - 1 = 5 > 0 \]
- For \( x = 2 \):
\[ f'(2) = 6(2)^5 - 1 = 6(32) - 1 = 192 - 1 = 191 > 0 \]

Since \( f'(x) \) is positive for both \( x = 1 \) and \( x = 2 \), and \( 6x^5 - 1 \) is a continuous function, we can conclude that \( f'(x) > 0 \) for all \( x \) in the interval \((1, 2)\).

**Hint:** If the derivative is positive over an interval, the function is increasing on that interval.

### Step 3: Determine if \( f(x) \) has a root in \([1, 2]\)
Next, we evaluate \( f(x) \) at the endpoints:
- For \( x = 1 \):
\[ f(1) = 1^6 - 1 - 1 = 1 - 1 - 1 = -1 \]
- For \( x = 2 \):
\[ f(2) = 2^6 - 2 - 1 = 64 - 2 - 1 = 61 \]

Now we have:
- \( f(1) = -1 \) (negative)
- \( f(2) = 61 \) (positive)

Since \( f(1) < 0 \) and \( f(2) > 0 \), and because \( f(x) \) is continuous (as it is a polynomial), by the Intermediate Value Theorem, there must be at least one root in the interval \((1, 2)\).

**Hint:** The Intermediate Value Theorem states that if a continuous function takes on opposite signs at two points, there is at least one root in between.

### Step 4: Conclusion about the statements
From our analysis:
- **Statement A:** \( f \) is increasing on \([1, 2]\) - **True**
- **Statement B:** \( f \) has a root in \([1, 2]\) - **True**
- **Statement C:** \( f \) is decreasing on \([1, 2]\) - **False**
- **Statement D:** \( f \) has no root in \([1, 2]\) - **False**

Thus, the correct statements are A and B.

**Final Answer:**
- A: True
- B: True
- C: False
- D: False