The range of `f(x)=(x+1)(x+2)(x+3)(x+4)+5` for `x in [-6,6]` is [4, 5045] (b) [0, 5045] `[-20 , 5045]` (d) none of these

Options

A. [4, 5045]
B. [0, 5045]
C. [-20, 5045]
D. None of these

Explanation

To find the range of the function \( f(x) = (x+1)(x+2)(x+3)(x+4) + 5 \) for \( x \) in the interval \([-6, 6]\), we can follow these steps:

### Step 1: Simplify the Function
First, we can simplify the expression \( (x+1)(x+2)(x+3)(x+4) \).

Let:
\[ g(x) = (x+1)(x+2)(x+3)(x+4) \]

### Step 2: Expand \( g(x) \)
We can expand \( g(x) \):
\[ g(x) = [(x+1)(x+4)] \cdot [(x+2)(x+3)] \]
Calculating each part:
\[ (x+1)(x+4) = x^2 + 5x + 4 \]
\[ (x+2)(x+3) = x^2 + 5x + 6 \]
Now, multiply these two results:
\[ g(x) = (x^2 + 5x + 4)(x^2 + 5x + 6) \]

### Step 3: Substitute \( t = x^2 + 5x \)
Let \( t = x^2 + 5x \). Then we can rewrite \( g(x) \) as:
\[ g(x) = (t + 4)(t + 6) = t^2 + 10t + 24 \]

### Step 4: Find the Minimum Value of \( g(x) \)
To find the minimum value of \( g(x) \), we first need to find the minimum value of \( t \) in the interval \([-6, 6]\).

Calculate \( t \) at the endpoints:
- For \( x = -6 \):
\[ t = (-6)^2 + 5(-6) = 36 - 30 = 6 \]
- For \( x = 6 \):
\[ t = (6)^2 + 5(6) = 36 + 30 = 66 \]

Now, we also need to check the vertex of the quadratic \( x^2 + 5x \):
The vertex occurs at \( x = -\frac{b}{2a} = -\frac{5}{2} = -2.5 \):
\[ t = (-2.5)^2 + 5(-2.5) = 6.25 - 12.5 = -6.25 \]

### Step 5: Determine the Range of \( g(x) \)
The minimum value of \( t \) in the interval \([-6, 6]\) is \(-6.25\) and the maximum is \(66\).

Now substitute these values back into \( g(x) \):
- Minimum \( g(-6.25) = (-6.25 + 4)(-6.25 + 6) = (-2.25)(-0.25) = 0.5625 \)
- Maximum \( g(66) = (66 + 4)(66 + 6) = 70 \cdot 72 = 5040 \)

### Step 6: Add 5 to Find \( f(x) \)
Now, we calculate \( f(x) \):
\[ f(x) = g(x) + 5 \]
- Minimum value of \( f(x) = 0.5625 + 5 = 5.5625 \)
- Maximum value of \( f(x) = 5040 + 5 = 5045 \)

### Conclusion
Thus, the range of \( f(x) \) for \( x \in [-6, 6] \) is:
\[ [5.5625, 5045] \]

### Final Answer
The correct option is (d) none of these.