Q576: For real x, let $f(x)=x^{3}+5x+1$. Then:...

Class CBSE Class 12 Mathematics Relations and Functions Q #576

KNOWLEDGE BASED

UNDERSTAND

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

For real x, let $f(x)=x^{3}+5x+1$. Then:

(A) f is one-one but not onto on R

(B) f is onto on R but not one-one

(D) f is neither one-one nor onto on R

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Correct Answer: C

AI Tutor Explanation

Detailed Solution

Step 1: Check for one-one nature

To check if the function is one-one, we need to analyze its derivative. If the derivative is always positive or always negative, the function is one-one.

Step 2: Find the derivative of f(x)

The derivative of $f(x) = x^3 + 5x + 1$ is $f'(x) = 3x^2 + 5$.

Step 3: Analyze the derivative

Since $x^2$ is always non-negative for real x, $3x^2$ is also non-negative. Therefore, $3x^2 + 5$ is always greater than or equal to 5, which means $f'(x) > 0$ for all real x. This implies that the function is strictly increasing.

Step 4: Conclude about one-one nature

Since the function is strictly increasing, it is one-one.

Step 5: Check for onto nature

To check if the function is onto, we need to see if the range of the function is equal to the codomain (which is R in this case). Since $f(x)$ is a polynomial of odd degree, its range is R.

Step 6: Conclude about onto nature

As $x$ approaches $-\infty$, $f(x)$ approaches $-\infty$, and as $x$ approaches $\infty$, $f(x)$ approaches $\infty$. Also, since $f(x)$ is continuous, it takes on all real values. Therefore, the function is onto.

Step 7: Final Conclusion

The function $f(x)$ is both one-one and onto on R.

Final Answer: f is one-one and onto on R

AI Suggestion: Option C

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concepts of one-one and onto functions and apply calculus to determine the nature of the given function.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concepts of functions, derivatives, one-one, and onto. It's not just recalling facts but applying these concepts.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of functions and calculus, which are core concepts in the syllabus.

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