The function `f(x)=2x^3-3x^2-12x+4` has

Options

A. no maxima and minima
B. one maximum and one minimum
C. two maximum
D. two minima

Answer

To solve the problem regarding the function \( f(x) = 2x^3 - 3x^2 - 12x + 4 \) and to determine the number of maxima and minima, we will follow these steps:

### Step 1: Find the first derivative of the function
We start by differentiating the function \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 12x + 4) \]

Calculating the derivative:

\[ f'(x) = 6x^2 - 6x - 12 \]

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

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

### Step 3: Simplify the equation
We can simplify this equation by dividing all terms by 6:

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

### Step 4: Factor the quadratic equation
Next, we factor the quadratic equation:

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

### Step 5: Solve for critical points
Setting each factor to zero gives us the critical points:

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

### Step 6: Find the second derivative
Now we need to determine the nature of these critical points by finding the second derivative:

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

Calculating the second derivative:

\[ f''(x) = 12x - 6 \]

### Step 7: Evaluate the second derivative at critical points
We evaluate the second derivative at the critical points \( x = 2 \) and \( x = -1 \):

1. For \( x = 2 \):

\[ f''(2) = 12(2) - 6 = 24 - 6 = 18 \quad (\text{which is } > 0) \]

This indicates that there is a **local minimum** at \( x = 2 \).

2. For \( x = -1 \):

\[ f''(-1) = 12(-1) - 6 = -12 - 6 = -18 \quad (\text{which is } < 0) \]

This indicates that there is a **local maximum** at \( x = -1 \).

### Conclusion
From our analysis, we have found:
- 1 local maximum at \( x = -1 \)
- 1 local minimum at \( x = 2 \)

Thus, the function \( f(x) = 2x^3 - 3x^2 - 12x + 4 \) has **1 maximum and 1 minimum**.

### Final Answer
The correct option is **B: 1 maxima and 1 minima**.

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