Calculating Horizontal Tangents for f(x)

Problem Statement

• Find the x-coordinate of all points on the graph of the function
  $f(x) = x^3 - x^2 - x + 5$
  at which the tangent line is horizontal.

Understanding Horizontal Tangent Lines

• A tangent line to a curve is horizontal when the slope of the curve at that point is zero.
• To find the slope of the curve, we need to determine the first derivative of the function, $f'(x)$.

Finding the Derivative of the Function

• We compute the derivative of the function using the power rule, which states that the derivative of $x^n$ is $n \cdot x^{n-1}$.
• Therefore,
  $f'(x) = 3x^2 - 2x - 1$.

Setting the Derivative to Zero

• To find the x-coordinates where the tangent line is horizontal, we set the derivative equal to zero:
  $3x^2 - 2x - 1 = 0$.

Solving the Quadratic Equation

• We will apply the quadratic formula where for a general equation $ax^2 + bx + c = 0$, the roots are given by:
  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• In our equation:

  • $a = 3$
  • $b = -2$
  • $c = -1$

• Plugging in these values into the quadratic formula:
  $x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3}$

  • Calculating the discriminant:
  • $(-2)^2 = 4$
  • $-4 \cdot 3 \cdot -1 = 12$
  • Therefore,
    $4 + 12 = 16$
  • Now substituting back:
    $x = \frac{2 \pm \sqrt{16}}{6}$
  • Which simplifies to:
    $x = \frac{2 \pm 4}{6}$
  • This gives us two possible solutions:
  1. $x = \frac{6}{6} = 1$
  2. $x = \frac{-2}{6} = -\frac{1}{3}$.

Summary of Solutions

• The x-coordinates where the tangent line is horizontal are:
  • (A) $x = 1$
  • (C) $x = -\frac{1}{3}$

Answer Choices Analysis

• The answer choices provided in the problem statement:

  • (A) $x = 1$
  • (B) $x = -1$
  • (C) $x = \frac{1}{3}$
  • (D) $x = -3, 1$
  • (E) $x = 1, -1$

• The correct answers based on the calculations are:

  • (A) $x = 1$
  • The choice of $x = -\frac{1}{3}$ does not match any of the options listed above, indicating that there may be an error or omission in the answer choices.

Conclusion

• The solutions indicate that the points on the graph of the function where the tangent line is horizontal are at
  • $x = 1$ and $x = -\frac{1}{3}$.
• Further steps should include verifying if any additional context or adjustments in the answer formatting is needed to check against provided answer choices.