The number of inflection points of a function given by a third degree polynomial is exactly

Options

A. 2
B. 1
C. 3
D. 0

Answer

To determine the number of inflection points of a third-degree polynomial, we can follow these steps:

### Step 1: Define the polynomial function
Let the third-degree polynomial be represented as:
\[ f(x) = ax^3 + bx^2 + cx + d \]
where \( a, b, c, \) and \( d \) are constants.

**Hint:** Identify the general form of a third-degree polynomial.

### Step 2: Find the first derivative
To find the inflection points, we first need to calculate the first derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(ax^3 + bx^2 + cx + d) \]
Using the power rule, we get:
\[ f'(x) = 3ax^2 + 2bx + c \]

**Hint:** Use the power rule for differentiation to find the first derivative.

### Step 3: Find the second derivative
Next, we find the second derivative \( f''(x) \):
\[ f''(x) = \frac{d}{dx}(3ax^2 + 2bx + c) \]
Again applying the power rule, we have:
\[ f''(x) = 6ax + 2b \]

**Hint:** Differentiate the first derivative to obtain the second derivative.

### Step 4: Set the second derivative to zero
To find the inflection points, we set the second derivative equal to zero:
\[ 6ax + 2b = 0 \]

**Hint:** Setting the second derivative to zero helps us find potential inflection points.

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ 6ax = -2b \]
\[ x = -\frac{b}{3a} \]

**Hint:** Isolate \( x \) to find the value at which the second derivative is zero.

### Step 6: Determine the number of inflection points
Since \( f''(x) \) is a linear function (as it is a first-degree polynomial in \( x \)), it can change sign at most once. Thus, there is exactly one inflection point at \( x = -\frac{b}{3a} \).

**Hint:** A linear function can only cross the x-axis once, indicating one inflection point.

### Conclusion
The number of inflection points of a third-degree polynomial is exactly **one**.

**Final Answer:** 1 inflection point.

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