The critical points of the function `y = (3)/(2)x^(4) - 4x^(3) - 45x^(2) + 51` are :

Options

A. `-5, -3, 0`
B. 0, 3, 5
C. `-5, 0, 3`
D. `-3, 0, 5`

Answer

To find the critical points of the function \( y = \frac{3}{2}x^4 - 4x^3 - 45x^2 + 51 \), we need to follow these steps:

### Step 1: Differentiate the function
We start by finding the first derivative of the function with respect to \( x \).

\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{3}{2}x^4 - 4x^3 - 45x^2 + 51 \right) \]

Using the power rule for differentiation:

\[ \frac{dy}{dx} = 4 \cdot \frac{3}{2}x^{4-1} - 3 \cdot 4x^{3-1} - 2 \cdot 45x^{2-1} + 0 \]

This simplifies to:

\[ \frac{dy}{dx} = 6x^3 - 12x^2 - 90x \]

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

\[ 6x^3 - 12x^2 - 90x = 0 \]

### Step 3: Factor the equation
We can factor out the common term \( 6x \):

\[ 6x(x^2 - 2x - 15) = 0 \]

Now, we can set each factor to zero:

1. \( 6x = 0 \)
2. \( x^2 - 2x - 15 = 0 \)

From the first factor, we get:

\[ x = 0 \]

### Step 4: Solve the quadratic equation
Next, we solve the quadratic equation \( x^2 - 2x - 15 = 0 \) using the factorization method:

\[ x^2 - 5x + 3x - 15 = 0 \]

This can be factored as:

\[ (x - 5)(x + 3) = 0 \]

Setting each factor to zero gives us:

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

### Step 5: List the critical points
Combining all the values we found, the critical points are:

\[ x = 0, \quad x = 5, \quad x = -3 \]

Thus, the critical points of the function are \( -3, 0, 5 \).

### Final Answer
The critical points of the function are \( -3, 0, 5 \).

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