f (x) is an even periodic function with period 10 in `[0,5], f (x) = {{:(2x, 0le x lt2),(3x ^(2)-8,2 le x lt 4),(10x, 4 le x le 5):}.` Then:

Options

A. `f (-4) =40`
B. `(f(-13) -f(11))/(f (13) +f(-11))=17/21`
C. `f (5)` is not defined
D. Range of `f (x)` is `[0,50]`

Answer

To solve the problem step by step, we need to analyze the given function \( f(x) \) defined on the intervals and check the conditions of being an even periodic function with a period of 10.

### Step 1: Understand the function definition
The function \( f(x) \) is defined as:
- \( f(x) = 2x \) for \( 0 \leq x < 2 \)
- \( f(x) = 3x^2 - 8 \) for \( 2 \leq x < 4 \)
- \( f(x) = 10x \) for \( 4 \leq x \leq 5 \)

### Step 2: Check if \( f(-x) = f(x) \) (Even Function)
Since \( f(x) \) is given to be an even function, we know that \( f(-x) = f(x) \).

#### For \( x = -4 \):
- \( f(-4) = f(4) \) (by even property)
- For \( x = 4 \), \( f(4) = 10 \times 4 = 40 \)
- Thus, \( f(-4) = 40 \)

### Step 3: Check the periodicity
The function is periodic with a period of 10. This means:
- \( f(x + 10) = f(x) \) for all \( x \)

### Step 4: Evaluate \( f(-13) - f(11) \)
Using the periodic property:
- \( f(-13) = f(7) \) (since \(-13 + 10 = -3\) and \(-3 + 10 = 7\))
- \( f(11) = f(1) \) (since \(11 - 10 = 1\))

Now we need to evaluate \( f(7) \) and \( f(1) \):
- For \( x = 7 \): \( f(7) = f(7 - 10) = f(-3) = f(3) \) (by even property)
- For \( x = 3 \): \( f(3) = 3(3^2) - 8 = 27 - 8 = 19 \)
- For \( x = 1 \): \( f(1) = 2(1) = 2 \)

Now substituting back:
\[ f(-13) - f(11) = f(7) - f(1) = 19 - 2 = 17 \]

### Step 5: Evaluate \( f(5) \)
From the function definition:
- \( f(5) = 10 \times 5 = 50 \)
- Thus, \( f(5) \) is defined.

### Step 6: Determine the range of \( f(x) \)
We need to check the values of \( f(x) \) across all intervals:
1. For \( 0 \leq x < 2 \):
- Minimum at \( x = 0 \): \( f(0) = 0 \)
- Maximum at \( x = 2 \): \( f(2) = 4 \)
- Range: \( [0, 4] \)

2. For \( 2 \leq x < 4 \):
- At \( x = 2 \): \( f(2) = 4 \)
- At \( x = 4 \): \( f(4) = 40 \)
- The function \( 3x^2 - 8 \) is a parabola opening upwards, so it will attain values from \( 4 \) to \( 40 \).
- Range: \( [4, 40] \)

3. For \( 4 \leq x \leq 5 \):
- At \( x = 4 \): \( f(4) = 40 \)
- At \( x = 5 \): \( f(5) = 50 \)
- Range: \( [40, 50] \)

### Final Range
Combining all ranges:
- Minimum value is \( 0 \) and maximum value is \( 50 \).
- Thus, the overall range is \( [0, 50] \).

### Conclusion
The correct options based on the analysis are:
1. \( f(-4) = 40 \) (Correct)
2. \( f(-13) - f(11) = 17 \) (Correct)
3. \( f(5) \) is defined (Correct)
4. The range of \( f(x) \) is \( [0, 50] \) (Correct)