The solution set of the inequality ` (x+3)^(5) -(x -1)^(5) ge 244` is

Options

A. `( -oo, 2]`
B. `[0, oo)`
C. `(-2,-1)`
D. (0,1)

Answer

To solve the inequality \( (x+3)^{5} - (x-1)^{5} \geq 244 \), we can follow these steps:

### Step 1: Rewrite the inequality
We start by rewriting the number 244 in a form that relates to powers of 3 and -1:
\[ 244 = 243 + 1 = 3^5 + (-1)^5 \]
Thus, we can rewrite the inequality as:
\[ (x+3)^{5} - (x-1)^{5} \geq 3^5 + (-1)^5 \]

### Step 2: Analyze the function
We need to analyze the function \( f(x) = (x+3)^{5} - (x-1)^{5} \). To understand its behavior, we can evaluate it at a specific point, such as \( x = 0 \):
\[ f(0) = (0+3)^{5} - (0-1)^{5} = 3^{5} - (-1)^{5} = 243 + 1 = 244 \]
This shows that \( f(0) = 244 \).

### Step 3: Determine the nature of the function
Next, we need to determine whether \( f(x) \) is increasing or decreasing. The function \( f(x) \) is a polynomial of degree 5, and since the leading coefficient is positive, it is an increasing function for all \( x \).

### Step 4: Solve the inequality
Since \( f(0) = 244 \) and \( f(x) \) is increasing, we can conclude that:
\[ f(x) \geq 244 \quad \text{for} \quad x \geq 0 \]
Thus, the solution set for the inequality \( (x+3)^{5} - (x-1)^{5} \geq 244 \) is:
\[ x \geq 0 \]

### Final Answer
The solution set is:
\[ [0, \infty) \]