Find the number of critical points of the function
`f (x) = - (3)/(4) x^(4) - 8x^(3) - (45)/(2) x^(2) + 105 , x in R`

Answer

To find the number of critical points of the function \( f(x) = -\frac{3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 105 \), we need to follow these steps:

### Step 1: Find the derivative of the function
To find the critical points, we first need to compute the first derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}\left(-\frac{3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 105\right) \]

Using the power rule for differentiation, we get:

\[ f'(x) = -3x^3 - 24x^2 - 45x \]

### Step 2: Set the derivative equal to zero
Next, we set the derivative equal to zero to find the critical points:

\[ -3x^3 - 24x^2 - 45x = 0 \]

### Step 3: Factor the derivative
We can factor out a common term from the equation:

\[ -3x(x^2 + 8x + 15) = 0 \]

This gives us one critical point at \( x = 0 \).

### Step 4: Factor the quadratic
Now, we need to factor the quadratic \( x^2 + 8x + 15 \):

\[ x^2 + 8x + 15 = (x + 3)(x + 5) \]

### Step 5: Find all critical points
Setting each factor equal to zero gives us:

1. \( -3x = 0 \) → \( x = 0 \)
2. \( x + 3 = 0 \) → \( x = -3 \)
3. \( x + 5 = 0 \) → \( x = -5 \)

Thus, the critical points are \( x = 0, -3, -5 \).

### Step 6: Count the number of critical points
We have found three critical points: \( x = 0, -3, -5 \).

### Conclusion
The number of critical points of the function \( f(x) \) is **3**.

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