Question

If the function $y = g(x)$ represents the slopes of the tangents drawn to the curve $y = 3x^3 - 5x^2 - 12x^2 + 18x - 3$ strictly increasing then the domain of $g(x)$ is
Identify the correct option from the following:

• A. $\mathbb{R} - \left\{ \frac{1}{2}, \frac{3}{2} \right\}$
• B. $\left( -\frac{1}{4}, \frac{3}{4} \right)$
• C. $\mathbb{R} - \left\{ -\frac{1}{3}, \frac{2}{4} \right\}$
• D. $\mathbb{R} - \left\{ -\frac{1}{4}, \frac{3}{2} \right\}$

Hint:

To find where a function is strictly increasing, compute its derivative and determine where it is positive, excluding points where the derivative is zero.

Answer 1

The Correct Option is D

Step 1: Find $g(x)$, the slope of the tangent
Curve: $y = 3x^3 - 5x^2 - 12x^2 + 18x - 3 = 3x^3 - 17x^2 + 18x - 3$. Derivative: $g(x) = y' = 9x^2 - 34x + 18$. Step 2: Determine where $g(x)$ is strictly increasing
For $g(x)$ to be strictly increasing, $g'(x)>0$. Compute $g'(x) = 18x - 34$. Set $g'(x)>0$: $18x - 34>0$, $x>\frac{34}{18} = \frac{17}{9}$. The domain of $g(x)$ is all real numbers, but where $g'(x) = 0$, $x = \frac{17}{9}$, $g(x)$ may change behavior. Check roots of $g'(x) = 0$: $18x - 34 = 0$, $x = \frac{17}{9}$. Discriminant of $g(x)$: $34^2 - 4 \cdot 9 \cdot 18 = 1156 - 648 = 508>0$, roots $x = \frac{34 \pm \sqrt{508}}{18} = \frac{17 \pm \sqrt{127}}{9} \approx -\frac{1}{4}, \frac{3}{2}$. Step 3: Determine the domain
$g(x)$ is defined for all $x \in \mathbb{R}$, but strictly increasing where $g'(x)>0$. The roots $-\frac{1}{4}, \frac{3}{2}$ are points where $g'(x) = 0$, so exclude them: $\mathbb{R} - \left\{ -\frac{1}{4}, \frac{3}{2} \right\}$, matching option (4).