The largest value of `2x^(3)-3x^(2)-12x+5` for `-2 le x le 2` occurs when

Options

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

Answer

To find the largest value of the function \( f(x) = 2x^3 - 3x^2 - 12x + 5 \) in the interval \([-2, 2]\), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 12x + 5) = 6x^2 - 6x - 12 \]

**Hint:** Remember that the derivative gives us the slope of the function, and we need to find critical points where this slope is zero.

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

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

Dividing the entire equation by 6 simplifies it:

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

**Hint:** This is a quadratic equation, and you can solve it using factoring or the quadratic formula.

### Step 3: Factor the quadratic equation
We factor the quadratic equation:

\[ x^2 - 2x + x - 2 = 0 \]

Grouping the terms gives us:

\[ (x - 2)(x + 1) = 0 \]

**Hint:** Factoring helps us find the roots of the equation easily.

### Step 4: Solve for \( x \)
Setting each factor to zero gives us the critical points:

\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

**Hint:** Make sure to check if these critical points are within the given interval \([-2, 2]\).

### Step 5: Evaluate the function at critical points and endpoints
Now we will evaluate the function at the critical points \( x = -1 \) and \( x = 2 \), as well as at the endpoints of the interval \( x = -2 \) and \( x = 2 \).

1. \( f(-2) = 2(-2)^3 - 3(-2)^2 - 12(-2) + 5 = -16 - 12 + 24 + 5 = 1 \)
2. \( f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + 5 = -2 - 3 + 12 + 5 = 12 \)
3. \( f(2) = 2(2)^3 - 3(2)^2 - 12(2) + 5 = 16 - 12 - 24 + 5 = -15 \)

**Hint:** Evaluating the function at these points helps us determine the maximum value.

### Step 6: Compare the values
Now we compare the values obtained:

- \( f(-2) = 1 \)
- \( f(-1) = 12 \)
- \( f(2) = -15 \)

The largest value occurs at \( x = -1 \).

### Conclusion
Thus, the largest value of \( f(x) \) in the interval \([-2, 2]\) occurs when \( x = -1 \).

**Final Answer:** The largest value occurs when \( x = -1 \).

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