If the equation `4x^(3)+5x+k=0(k in R)` has a negative real root then (a) `k=0` (b) `-ooltklt0` (c) `0ltkltoo` (d) `-ooltkltoo`

Options

A. k=0
B. `-ooltklt0`
C. `0ltkltoo`
D. `-ooltkltoo`

Explanation

To solve the problem, we need to analyze the cubic equation given by \( f(x) = 4x^3 + 5x + k \) and determine the conditions under which it has a negative real root.

### Step 1: Understand the function and its behavior
The function \( f(x) = 4x^3 + 5x + k \) is a cubic polynomial. We need to find the conditions on \( k \) such that this function has a negative root.

### Step 2: Find the derivative of the function
To analyze the behavior of the function, we first find its derivative:
\[ f'(x) = \frac{d}{dx}(4x^3 + 5x + k) = 12x^2 + 5 \]

### Step 3: Analyze the derivative
The derivative \( f'(x) = 12x^2 + 5 \) is always positive for all \( x \in \mathbb{R} \) since \( 12x^2 \) is non-negative and \( 5 \) is positive. This means that \( f(x) \) is a strictly increasing function.

### Step 4: Determine the implications of a strictly increasing function
Since \( f(x) \) is strictly increasing, it can cross the x-axis at most once. If it has a negative root, it must cross the x-axis from above to below. Therefore, we need to evaluate \( f(0) \):
\[ f(0) = 4(0)^3 + 5(0) + k = k \]
For \( f(x) \) to have a negative root, we require:
\[ f(0) > 0 \implies k > 0 \]

### Step 5: Conclusion about the value of \( k \)
Since \( k \) must be greater than 0 for the function to have a negative root, we conclude:
\[ 0 < k < \infty \]

Thus, the correct answer is:
(c) \( 0 < k < \infty \)