For what value of a, does the function `f (x) = x ^(5) - 5x + a` has three real roots ?

Options

A. `a gt 4`
B. `a lt 4`
C. `-4 lt a lt 4`
D. `-4 le a le 4`

Explanation

To find the value of \( a \) such that the function \( f(x) = x^5 - 5x + a \) has three real roots, we can follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^5 - 5x + a) = 5x^4 - 5 \]

### Step 2: Set the derivative to zero
Next, we set the derivative equal to zero to find the critical points:
\[ 5x^4 - 5 = 0 \]
\[ x^4 = 1 \]
\[ x = 1 \quad \text{or} \quad x = -1 \]

### Step 3: Analyze the critical points
The critical points are \( x = 1 \) and \( x = -1 \). We need to evaluate the function \( f(x) \) at these points to determine the nature of the roots.

### Step 4: Evaluate \( f(x) \) at critical points
Calculate \( f(-1) \) and \( f(1) \):
\[ f(-1) = (-1)^5 - 5(-1) + a = -1 + 5 + a = a + 4 \]
\[ f(1) = (1)^5 - 5(1) + a = 1 - 5 + a = a - 4 \]

### Step 5: Apply the Intermediate Value Theorem
For the function to have three real roots, it must change signs around the critical points. Therefore, we need:
1. \( f(-1) > 0 \) (which implies \( a + 4 > 0 \) or \( a > -4 \))
2. \( f(1) < 0 \) (which implies \( a - 4 < 0 \) or \( a < 4 \))

### Step 6: Combine the inequalities
Combining the inequalities from Step 5 gives us:
\[ -4 < a < 4 \]

### Conclusion
Thus, the value of \( a \) for which the function \( f(x) = x^5 - 5x + a \) has three real roots is:
\[ a \in (-4, 4) \]