Let `f(x)=30-2x -x^3`, the number of the positive integral values of `x` which does satisfy `f(f(f(x)))) gt f(f(-x))` is _______.

Explanation

To solve the problem, we need to analyze the function \( f(x) = 30 - 2x - x^3 \) and determine the number of positive integral values of \( x \) that satisfy the inequality \( f(f(f(x))) > f(f(-x)) \).

### Step 1: Analyze the function \( f(x) \)

1. **Find the derivative \( f'(x) \)**:
\[ f'(x) = -2 - 3x^2 \]
Since \( -2 - 3x^2 < 0 \) for all \( x \), the function \( f(x) \) is decreasing for all \( x \).

**Hint**: Check the sign of the derivative to determine if the function is increasing or decreasing.

### Step 2: Understand the implications of \( f(x) \) being decreasing

Since \( f(x) \) is a decreasing function, we can conclude that:
- If \( x_1 > x_2 \), then \( f(x_1) < f(x_2) \).

**Hint**: Remember that for decreasing functions, larger inputs yield smaller outputs.

### Step 3: Set up the inequality

We need to analyze the inequality:
\[ f(f(f(x))) > f(f(-x)) \]
Given that \( f \) is decreasing, we can rewrite this as:
\[ f(f(x)) < f(-x) \]
And further:
\[ f(x) > -x \]

**Hint**: Use the property of decreasing functions to reverse the inequality when you apply \( f \).

### Step 4: Solve the inequality \( f(x) > -x \)

Substituting \( f(x) \):
\[ 30 - 2x - x^3 > -x \]
Rearranging gives:
\[ 30 - 2x + x - x^3 > 0 \]
This simplifies to:
\[ -x^3 - x + 30 > 0 \]
or
\[ x^3 + x - 30 < 0 \]

**Hint**: Rearranging the inequality can help you identify the roots of the polynomial.

### Step 5: Find the roots of the cubic equation

To find the roots of \( x^3 + x - 30 = 0 \), we can test some integer values:
- For \( x = 3 \):
\[ 3^3 + 3 - 30 = 27 + 3 - 30 = 0 \]
Thus, \( x = 3 \) is a root.

Now, we can factor the cubic polynomial:
\[ x^3 + x - 30 = (x - 3)(x^2 + 3x + 10) \]

### Step 6: Analyze the quadratic factor

The quadratic \( x^2 + 3x + 10 \) has a discriminant:
\[ D = 3^2 - 4 \cdot 1 \cdot 10 = 9 - 40 = -31 \]
Since the discriminant is negative, \( x^2 + 3x + 10 > 0 \) for all \( x \).

Thus, the sign of \( x^3 + x - 30 \) is determined by \( x - 3 \):
- \( x^3 + x - 30 < 0 \) for \( x < 3 \).

### Step 7: Find positive integral values of \( x \)

The positive integral values of \( x \) that satisfy \( x < 3 \) are:
- \( x = 1 \)
- \( x = 2 \)

Thus, the number of positive integral values of \( x \) that satisfy the original inequality is **2**.

### Final Answer
The number of positive integral values of \( x \) which satisfy \( f(f(f(x))) > f(f(-x)) \) is **2**.

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