Let `f (x) = x^3 + ax^2 + bx + 5 sin^2` x be an increasing function on the set R. Then a and b satisfy

Options

A. ` a^2 - 3b - 15 gt 0`
B. ` a^2 - 3b +15 gt 0`
C. ` a^2 - 3b + 15 lt 0`
D. ` a gt 0` and ` b gt 0`

Answer

To determine the conditions under which the function \( f(x) = x^3 + ax^2 + bx + 5 \sin^2 x \) is an increasing function on the set of real numbers \( \mathbb{R} \), we need to analyze its derivative.

### Step 1: Find the derivative of \( f(x) \)
The first step is to compute the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(ax^2) + \frac{d}{dx}(bx) + \frac{d}{dx}(5 \sin^2 x) \]
Calculating each term:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( ax^2 \) is \( 2ax \).
- The derivative of \( bx \) is \( b \).
- The derivative of \( 5 \sin^2 x \) using the chain rule is \( 5 \cdot 2 \sin x \cos x = 5 \sin(2x) \).

Thus, we have:
\[ f'(x) = 3x^2 + 2ax + b + 5 \sin(2x) \]

### Step 2: Set the condition for \( f'(x) \) to be non-negative
For \( f(x) \) to be an increasing function, we require:
\[ f'(x) \geq 0 \quad \text{for all } x \in \mathbb{R} \]
This means:
\[ 3x^2 + 2ax + b + 5 \sin(2x) \geq 0 \]

### Step 3: Analyze the term \( 5 \sin(2x) \)
The sine function oscillates between -1 and 1, so:
\[ -5 \leq 5 \sin(2x) \leq 5 \]
This implies that the minimum value of \( 5 \sin(2x) \) is -5. Therefore, we can rewrite our inequality:
\[ 3x^2 + 2ax + b - 5 \geq 0 \]

### Step 4: Formulate the quadratic inequality
We need to ensure that the quadratic \( 3x^2 + 2ax + (b - 5) \) is non-negative for all \( x \). This occurs if:
1. The leading coefficient is positive: \( 3 > 0 \) (which is true).
2. The discriminant of the quadratic is less than or equal to zero:
\[ D = (2a)^2 - 4 \cdot 3 \cdot (b - 5) \leq 0 \]

### Step 5: Solve the discriminant inequality
Calculating the discriminant:
\[ D = 4a^2 - 12(b - 5) \leq 0 \]
This simplifies to:
\[ 4a^2 - 12b + 60 \leq 0 \]
Dividing through by 4 gives:
\[ a^2 - 3b + 15 \leq 0 \]

### Conclusion
The conditions that \( a \) and \( b \) must satisfy for \( f(x) \) to be an increasing function on \( \mathbb{R} \) are:
\[ a^2 - 3b + 15 \leq 0 \]