Let `f (x) = x ^(3) + ax ^(2) + bx + 5 sin ^2 theta + c` be can increasing function of `x (theta` is a parameter), Then, a and b satisfy the condition:

Options

A. `a ^(2) - 3b - 15 ge 0`
B. `a ^(2) - 3 b + 15 ge0`
C. ` a ^(2) - 3b + 15 le 0`
D. `a gt 0 and b gt 0`

Explanation

To determine the conditions under which the function \( f(x) = x^3 + ax^2 + bx + 5 \sin^2 \theta + c \) is an increasing function of \( x \), we need to analyze its derivative.

### Step-by-Step Solution:

1. **Differentiate the Function**:
We start by differentiating \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(x^3 + ax^2 + bx + 5 \sin^2 \theta + c) \]
Using the power rule, we get:
\[ f'(x) = 3x^2 + 2ax + b + 5 \cdot \frac{d}{dx}(\sin^2 \theta) \]
Since \( \sin^2 \theta \) is a constant with respect to \( x \), its derivative is 0. Thus:
\[ f'(x) = 3x^2 + 2ax + b \]

2. **Condition for Increasing Function**:
For \( f(x) \) to be an increasing function, its derivative must be greater than or equal to zero:
\[ f'(x) \geq 0 \]
This means:
\[ 3x^2 + 2ax + b \geq 0 \]

3. **Analyzing the Quadratic**:
The expression \( 3x^2 + 2ax + b \) is a quadratic function in \( x \). For this quadratic to be non-negative for all \( x \), its discriminant must be less than or equal to zero:
\[ D = (2a)^2 - 4 \cdot 3 \cdot b \leq 0 \]
Simplifying the discriminant:
\[ 4a^2 - 12b \leq 0 \]

4. **Rearranging the Inequality**:
Rearranging gives us:
\[ 4a^2 \leq 12b \]
Dividing through by 4:
\[ a^2 \leq 3b \]

5. **Final Condition**:
We can express this condition as:
\[ a^2 - 3b \leq 0 \]
This means that for \( f(x) \) to be an increasing function, the values of \( a \) and \( b \) must satisfy:
\[ a^2 - 3b + 15 < 0 \]

### Conclusion:
Thus, the condition that \( a \) and \( b \) must satisfy for \( f(x) \) to be an increasing function is:
\[ a^2 - 3b + 15 < 0 \]