Let `f(x) = (3)/(4) x^(4) -x^(3) -9x^(2) + 7`, then the number of critical points in `[-1, 4]` is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find the number of critical points of the function \( f(x) = \frac{3}{4} x^4 - x^3 - 9x^2 + 7 \) in the interval \([-1, 4]\), we will follow these steps:

### Step 1: Differentiate the function
We need to find the first derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx} \left( \frac{3}{4} x^4 - x^3 - 9x^2 + 7 \right) \]

Using the power rule for differentiation, we get:

\[ f'(x) = 3x^3 - 3x^2 - 18x \]

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

\[ 3x^3 - 3x^2 - 18x = 0 \]

### Step 3: Factor the derivative
We can factor out the common term \( 3x \):

\[ 3x(x^2 - x - 6) = 0 \]

This gives us:

\[ 3x = 0 \quad \text{or} \quad x^2 - x - 6 = 0 \]

### Step 4: Solve for \( x \)
From \( 3x = 0 \), we get:

\[ x = 0 \]

Now we solve the quadratic equation \( x^2 - x - 6 = 0 \) using the factorization method:

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

This gives us:

\[ x = 3 \quad \text{or} \quad x = -2 \]

### Step 5: Identify critical points in the interval
Now we have the critical points \( x = 0, 3, -2 \). We need to check which of these points lie within the interval \([-1, 4]\):

- \( x = 0 \) is in the interval.
- \( x = 3 \) is in the interval.
- \( x = -2 \) is not in the interval.

### Step 6: Count the critical points
Thus, the critical points in the interval \([-1, 4]\) are \( x = 0 \) and \( x = 3 \).

### Conclusion
The number of critical points in the interval \([-1, 4]\) is **2**.

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