f(x)=x^3+kx^2+5x+4sin^2x be an increasing function on x€R then domain of k is:

A.(-√2,√2)

B.(-infinity,√3)

C.(√3,infinity)

D.(-√3,√3)

To determine the values of \( k \) for which the function \( f(x) = x^3 + kx^2 + 5x + 4\sin^2(x) \) is increasing for all \( x \in \mathbb{R} \), we need to analyze the derivative of the function. A function is increasing if its derivative is non-negative for all \( x \). Let's go through the steps to find the appropriate domain for \( k \).

Finding the Derivative

The first step is to compute the derivative of the function:

Given:

\( f(x) = x^3 + kx^2 + 5x + 4\sin^2(x) \)

We differentiate \( f(x) \) with respect to \( x \):

\( f'(x) = 3x^2 + 2kx + 5 + 8\sin(x)\cos(x) \)

Using the identity \( \sin(2x) = 2\sin(x)\cos(x) \), we can rewrite the derivative as:

\( f'(x) = 3x^2 + 2kx + 5 + 4\sin(2x) \)

Analyzing the Derivative

For \( f(x) \) to be increasing, we require \( f'(x) \geq 0 \) for all \( x \). This means we need to analyze the expression:

\( 3x^2 + 2kx + 5 + 4\sin(2x) \geq 0 \)

Since \( \sin(2x) \) oscillates between -1 and 1, the term \( 4\sin(2x) \) will oscillate between -4 and 4. Therefore, we can consider the worst-case scenario where \( \sin(2x) = -1 \):

In this case, we have:

\( 3x^2 + 2kx + 5 - 4 \geq 0 \)

which simplifies to:

\( 3x^2 + 2kx + 1 \geq 0 \)

Finding Conditions for Non-negativity

The quadratic \( 3x^2 + 2kx + 1 \) must be non-negative for all \( x \). For a quadratic \( ax^2 + bx + c \) to be non-negative for all \( x \), the following conditions must hold:

• The leading coefficient \( a \) must be positive (which it is, since \( a = 3 > 0 \)).
• The discriminant must be less than or equal to zero: \( b^2 - 4ac \leq 0 \).

In our case:

Here, \( a = 3 \), \( b = 2k \), and \( c = 1 \). The discriminant is:

\( (2k)^2 - 4(3)(1) \leq 0 \)

which simplifies to:

\( 4k^2 - 12 \leq 0 \)

or:

\( k^2 \leq 3 \)

Determining the Domain of k

Taking the square root of both sides gives us:

\( -\sqrt{3} \leq k \leq \sqrt{3} \)

This means that the values of \( k \) for which the function \( f(x) \) is increasing for all \( x \in \mathbb{R} \) are contained within the interval:

\( k \in (-\sqrt{3}, \sqrt{3}) \)

Final Answer

From the options provided, the correct choice is:

D. (-√3, √3)

This interval ensures that the quadratic component of the derivative remains non-negative, allowing the function to be increasing across its entire domain.