Choose a correct statement about the slope and equation of the tangent line to the curve

f(x)=x3−3x2−x+3  at the point  (x=2,y=−3).

A. No correct answer.

B.

Slope is −1 and equation is x−y+1=0

C.

Slope is −3 and equation is y=3−3x

D.

Slope is undefined and equation is x=2.

E.

Slope is 0 and equation is y=−3

To determine the correct statement about the slope and equation of the tangent line to the curve \( f(x) = x^3 - 3x^2 - x + 3 \) at the point \( (x = 2, y = -3) \), we need to:

1. Find the derivative of \( f(x) \), which gives us the slope of the tangent line at any point \( x \).
2. Evaluate the derivative at \( x = 2 \) to find the slope at that specific point.
3. Use the point-slope form of a line to find the equation of the tangent line.

### Step 1: Find the derivative of \( f(x) \)

The derivative of \( f(x) = x^3 - 3x^2 - x + 3 \) is:

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

\[
f'(x) = 3x^2 - 6x - 1
\]

### Step 2: Evaluate the derivative at \( x = 2 \)

Now, substitute \( x = 2 \) into \( f'(x) \) to find the slope:

\[
f'(2) = 3(2)^2 - 6(2) - 1
\]

\[
f'(2) = 3 \cdot 4 - 12 - 1
\]

\[
f'(2) = 12 - 12 - 1 = -1
\]

So, the slope of the tangent line at the point \( (2, -3) \) is \(-1\).

### Step 3: Find the equation of the tangent line

We use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \( (x_1, y_1) = (2, -3) \) and \( m = -1 \).

\[
y - (-3) = -1(x - 2)
\]

\[
y + 3 = -x + 2
\]

\[
y = -x + 2 - 3
\]

\[
y = -x - 1
\]

Rewriting this in standard form, we get:

\[
x + y + 1 = 0
\]

### Conclusion

The correct statement is:

**B. Slope is \(-1\) and the equation is \(x + y + 1 = 0\).**

Do you have any questions or would you like more details?

---

### Related Questions

1. How do you find the equation of a tangent line to a curve?
2. What is the geometric interpretation of the derivative of a function at a point?
3. How does the point-slope form of a line relate to finding tangents?
4. How would the problem change if the point of tangency were different?
5. What would the second derivative of \(f(x)\) indicate about the curve's concavity at \(x = 2\)?

**Tip:** To find the slope of a tangent line at a point, always calculate the derivative and evaluate it at the desired point.

Learn how to find the slope and equation of the tangent line to the curve f(x) = x^3 - 3x^2 - x + 3 at the point (x = 2, y = -3). Step-by-step solution includes calculating the derivative, evaluating the slope, and using the point-slope form of a line. Suitable for high school students studying calculus and tangent lines.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation