Q1290: Show that the function $f(x)=4x^{3}-18x^{2}+27x-7$ has neith...

CBSE Class 12 Mathematics Applications of Derivatives Q #1290

KNOWLEDGE BASED

UNDERSTAND

2 Marks 2024 AISSCE(Board Exam) VSA

Show that the function $f(x)=4x^{3}-18x^{2}+27x-7$ has neither maxima nor minima.

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AI Tutor Explanation

Detailed Solution

Step 1: Find the first derivative

To find the critical points, we first need to find the first derivative of the function $f(x)$. $$f'(x) = \frac{d}{dx}(4x^{3}-18x^{2}+27x-7) = 12x^{2} - 36x + 27$$

Step 2: Find the critical points

To find the critical points, we set the first derivative equal to zero and solve for $x$. $$12x^{2} - 36x + 27 = 0$$ Divide by 3: $$4x^{2} - 12x + 9 = 0$$ $$(2x - 3)^{2} = 0$$ $$2x - 3 = 0$$ $$x = \frac{3}{2}$$ So, there is only one critical point at $x = \frac{3}{2}$.

Step 3: Find the second derivative

To determine whether the critical point is a maximum, minimum, or neither, we find the second derivative of the function. $$f''(x) = \frac{d}{dx}(12x^{2} - 36x + 27) = 24x - 36$$

Step 4: Evaluate the second derivative at the critical point

Now, we evaluate the second derivative at the critical point $x = \frac{3}{2}$. $$f''\left(\frac{3}{2}\right) = 24\left(\frac{3}{2}\right) - 36 = 36 - 36 = 0$$ Since the second derivative is zero at the critical point, the second derivative test is inconclusive.

Step 5: Analyze the sign of the first derivative around the critical point

We need to analyze the sign of the first derivative around $x = \frac{3}{2}$. Since $f'(x) = (2x - 3)^{2}$, $f'(x)$ is always non-negative. For $x < \frac{3}{2}$, $f'(x) > 0$. For $x > \frac{3}{2}$, $f'(x) > 0$. Since the sign of the first derivative does not change around $x = \frac{3}{2}$, there is neither a maximum nor a minimum at this point. The function is increasing on both sides of $x = \frac{3}{2}$.

Step 6: Conclusion

Therefore, the function $f(x) = 4x^{3} - 18x^{2} + 27x - 7$ has neither maxima nor minima.

Final Answer: The function has neither maxima nor minima.

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to demonstrate understanding of how to use derivatives to determine maxima and minima and interpret the results.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure (finding first and second derivatives, setting them to zero, and interpreting the results) to determine if a function has maxima or minima.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests a standard procedure taught in the textbook for finding maxima and minima using derivatives.