Applications of Derivatives
Differential Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
$f : \mathbb{R} \to \mathbb{R}$
**Statement 1**: $f(x) = 12x^5 - 15x^4 + 20x^3 - 30x^2 + 60x + 1$ is monotonic and surjective on $\mathbb{R}$.
**Statement 2**: A continuous function defined on $\mathbb{R}$, if strictly monotonic has its range $\mathbb{R}$.
Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1
Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1
Statement-1 is true, statement-2 is false
Statement-1 is false, statement-2 is true
Step-by-Step Solution
Key Concept: Statement 1 is true by verifying $f'(x) > 0$ everywhere, but Statement 2's converse (surjective $\Rightarrow$ strictly monotonic) is false.
To verify Statement 1, compute $f'(x) = 60x^4 - 60x^3 + 60x^2 - 60x + 60 = 60(x^4 - x^3 + x^2 - x + 1)$. Factor this as $f'(x) = 60[(x^2 - rac{x}{2})^2 + rac{3x^2}{4} - x + 1]$, which simplifies to show $f'(x) > 0$ for all $x \in \mathbb{R}$ (the discriminant of the quadratic form is negative). Thus $f$ is strictly monotonic (increasing) and continuous on $\mathbb{R}$, making it surjective onto $\mathbb{R}$. Statement 1 is TRUE. Statement 2 is FALSE because it requires the converse implication: while a strictly monotonic continuous function has range $\mathbb{R}$, the converse is not always true—a function can be surjective without being monotonic.
Correct Answer: 3