Let `g(x)=int_0^x(3t^(2)+2t+9)dt and f(x) ` be a decreasing function `forallxge0` such that `AB=f(x)hat(i)+g(x)hat(j) and AC=g(x)hat(i)+f(x)hat(j)` are the two smallest sides of a triangle ABC whose circumcentre lies outside the triangle `forall cgt 0.` Q. Which of the following is true (for `xgeo)`

Options

A. (a)`f(x)gt0`, `g(x)lt0`
B. (b)`f(x)lt0`, `g(x)lt0`
C. (c)`f(x)gt0`, `g(x)gt0`
D. (d)`f(x)lt0`, `g(x)gt0`

Answer

To solve the problem step by step, we will analyze the given functions and the conditions provided in the question.

### Step 1: Define the Functions
We have:
- \( g(x) = \int_0^x (3t^2 + 2t + 9) \, dt \)
- \( f(x) \) is a decreasing function for all \( x \geq 0 \).

### Step 2: Compute \( g(x) \)
To compute \( g(x) \), we need to evaluate the integral:
\[ g(x) = \int_0^x (3t^2 + 2t + 9) \, dt \]
Calculating the integral:
\[ g(x) = \left[ t^3 + t^2 + 9t \right]_0^x = x^3 + x^2 + 9x \]

### Step 3: Find the Derivative of \( g(x) \)
Next, we find the derivative \( g'(x) \):
\[ g'(x) = \frac{d}{dx}(x^3 + x^2 + 9x) = 3x^2 + 2x + 9 \]
Since \( 3x^2 + 2x + 9 \) is a quadratic function with a positive leading coefficient, it is always positive for all \( x \geq 0 \). Thus, \( g(x) \) is an increasing function.

### Step 4: Analyze the Triangle Condition
We have two sides of the triangle:
- \( AB = f(x) \hat{i} + g(x) \hat{j} \)
- \( AC = g(x) \hat{i} + f(x) \hat{j} \)

For the circumcenter to lie outside the triangle, it must be an obtuse triangle. This occurs when one angle is greater than \( 90^\circ \).

### Step 5: Conditions for the Triangle
For the triangle to be obtuse, we need to analyze the lengths of the sides:
- The lengths of \( AB \) and \( AC \) can be expressed as:
\[ |AB| = \sqrt{(f(x))^2 + (g(x))^2} \]
\[ |AC| = \sqrt{(g(x))^2 + (f(x))^2} \]
Both lengths are equal, so we need to consider the angle between them.

### Step 6: Determine the Behavior of \( f(x) \)
Since \( f(x) \) is a decreasing function and \( g(x) \) is increasing, we can conclude:
- As \( x \) increases, \( g(x) \) increases while \( f(x) \) decreases.

### Step 7: Conclusion on the Options
Given the conditions:
- \( f(x) < 0 \) (since it is decreasing and must be negative for large \( x \))
- \( g(x) > 0 \) (since it is increasing and starts from 0)

Thus, the correct statement is:
- \( f(x) < 0 \) and \( g(x) > 0 \).

### Final Answer
The correct option is **D**: \( f(x) < 0 \) and \( g(x) > 0 \).