Q62 / 180 · COMEDK 2026 (9 May Shift 2) · Mathematics
If $f(x) = x^3 + \dfrac{3}{2}x^2 + 3x + 3$, then $f(x)$ is
- AOdd function
- BEven function
- Correct answerCIncreasing function
- DDecreasing function
Solution
Given function is $f(x) = x^3 + \dfrac{3}{2}x^2 + 3x + 3$.
Differentiating with respect to $x$, we get:
$f'(x) = 3x^2 + 3x + 3$
$f'(x) = 3(x^2 + x + 1)$
The discriminant of the quadratic $x^2 + x + 1$ is $D = 1^2 - 4(1)(1) = -3 < 0$.
Since the leading coefficient is positive and $D < 0$, $x^2 + x + 1 > 0$ for all $x \in \mathbb{R}$.
Therefore, $f'(x) > 0$ for all $x \in \mathbb{R}$.
This implies that $f(x)$ is an increasing function.
Answer: Increasing function
Differentiating with respect to $x$, we get:
$f'(x) = 3x^2 + 3x + 3$
$f'(x) = 3(x^2 + x + 1)$
The discriminant of the quadratic $x^2 + x + 1$ is $D = 1^2 - 4(1)(1) = -3 < 0$.
Since the leading coefficient is positive and $D < 0$, $x^2 + x + 1 > 0$ for all $x \in \mathbb{R}$.
Therefore, $f'(x) > 0$ for all $x \in \mathbb{R}$.
This implies that $f(x)$ is an increasing function.
Answer: Increasing function
Hence, the correct answer is option: 'C'.