The number of nonzero integral values of `a` for which the function `f(x)=x^4+a x^3+(3x^2)/2+1` is concave upward along the entire real line is___________

Explanation

To determine the number of nonzero integral values of \( a \) for which the function

\[ f(x) = x^4 + ax^3 + \frac{3x^2}{2} + 1 \]

is concave upward along the entire real line, we need to analyze the second derivative of the function.

### Step 1: Find the first derivative \( f'(x) \)

The first derivative of \( f(x) \) is calculated as follows:

\[ f'(x) = \frac{d}{dx}(x^4) + \frac{d}{dx}(ax^3) + \frac{d}{dx}\left(\frac{3x^2}{2}\right) + \frac{d}{dx}(1) \]

Calculating each term gives:

\[ f'(x) = 4x^3 + 3ax^2 + 3x \]

### Step 2: Find the second derivative \( f''(x) \)

Now, we differentiate \( f'(x) \) to find \( f''(x) \):

\[ f''(x) = \frac{d}{dx}(4x^3) + \frac{d}{dx}(3ax^2) + \frac{d}{dx}(3x) \]

Calculating each term gives:

\[ f''(x) = 12x^2 + 6ax + 3 \]

### Step 3: Set the condition for concavity

For the function to be concave upward along the entire real line, we need:

\[ f''(x) \geq 0 \quad \text{for all } x \]

This means that the quadratic \( 12x^2 + 6ax + 3 \) must not have any real roots, which occurs when its discriminant is less than or equal to zero.

### Step 4: Calculate the discriminant

The discriminant \( D \) of the quadratic \( Ax^2 + Bx + C \) is given by:

\[ D = B^2 - 4AC \]

For our quadratic \( 12x^2 + 6ax + 3 \):

- \( A = 12 \)
- \( B = 6a \)
- \( C = 3 \)

Thus, the discriminant is:

\[ D = (6a)^2 - 4 \cdot 12 \cdot 3 \]

Calculating this gives:

\[ D = 36a^2 - 144 \]

### Step 5: Set the discriminant less than or equal to zero

We need:

\[ 36a^2 - 144 \leq 0 \]

This simplifies to:

\[ 36a^2 \leq 144 \]

Dividing through by 36:

\[ a^2 \leq 4 \]

Taking the square root of both sides gives:

\[ -2 \leq a \leq 2 \]

### Step 6: Identify nonzero integral values of \( a \)

The integral values of \( a \) that satisfy this inequality are:

\[ -2, -1, 1, 2 \]

The nonzero integral values are:

\[ -2, -1, 1, 2 \]

### Conclusion

Thus, the number of nonzero integral values of \( a \) is:

\[ \boxed{4} \]