Question

The abscissa of the points, where the tangent to the curve $y = x^3 - 3x^2 - 9x + 5$ is parallel to $X$ axis are

• A. $x = 1$ and $-1$
• B. $x = 1$ and $-3$
• C. $x = -1$ and $3$
• D. $x = 0$ and $1$

Hint:

"Parallel to the $X$-axis" is a geometric code phrase for setting the derivative equal to zero ($\frac{dy}{dx} = 0$).
"Parallel to the $Y$-axis" means the tangent is vertical, so the derivative is undefined (set the denominator of $\frac{dy}{dx}$ equal to zero).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks for the abscissae (the $x$-coordinates) of the points on the given cubic curve where the tangent line is completely horizontal (parallel to the $X$-axis).

Step 2: Key Formula or Approach:
The slope of the tangent line to a curve $y = f(x)$ at any point is given by its first derivative, $\frac{dy}{dx}$.
When a line is parallel to the $X$-axis, its slope is equal to zero. Therefore, we must differentiate the equation, set $\frac{dy}{dx} = 0$, and solve the resulting equation for $x$.

Step 3: Detailed Explanation:
The equation of the curve is:
$$y = x^3 - 3x^2 - 9x + 5$$ Differentiating both sides with respect to $x$ using the power rule:
$$\frac{dy}{dx} = 3x^2 - 6x - 9$$ Setting the derivative equal to $0$ for a horizontal tangent:
$$3x^2 - 6x - 9 = 0$$ Divide the entire quadratic equation by $3$ to simplify:
$$x^2 - 2x - 3 = 0$$ Factor the quadratic equation by splitting the middle term:
$$x^2 - 3x + x - 3 = 0$$ $$x(x - 3) + 1(x - 3) = 0$$ $$(x - 3)(x + 1) = 0$$ This gives two possible solutions for the abscissa:
$$x = 3 \quad \text{or} \quad x = -1$$

Step 4: Final Answer:
The required abscissae are $x = -1$ and $3$, which corresponds to option (C).