Class CBSE Class 12 Mathematics Applications of Derivatives Q #1844
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
The least value of $f(x) = x^3 - 12x$, $x \in [0, 3]$ is
(A) $ -16$
(B) $ -9$
(C) $ 0$
(D) $ 16$
Prev Finish
Correct Answer: A

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Find the derivative

To find the critical points, we first differentiate the function $f(x) = x^3 - 12x$ with respect to $x$: $$f'(x) = 3x^2 - 12$$

Step 2: Identify critical points

Set $f'(x) = 0$ to find the critical points: $$3x^2 - 12 = 0$$ $$3(x^2 - 4) = 0$$ $$x^2 = 4 \implies x = 2, -2$$ Since the interval is $[0, 3]$, we only consider $x = 2$ as the critical point within the domain.

Step 3: Evaluate at critical points and endpoints

We evaluate $f(x)$ at the critical point $x = 2$ and the endpoints $x = 0$ and $x = 3$: $$f(0) = (0)^3 - 12(0) = 0$$ $$f(2) = (2)^3 - 12(2) = 8 - 24 = -16$$ $$f(3) = (3)^3 - 12(3) = 27 - 36 = -9$$

Step 4: Determine the least value

Comparing the values $0, -16,$ and $-9$, the minimum value is $-16$.

Final Answer: -16

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the First Derivative Test for local extrema and the Extreme Value Theorem for closed intervals.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step algorithmic approach involving differentiation, solving equations, and comparing values.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to handle optimization problems in calculus, a core component of the 'Applications of Derivatives' unit.

More from this Chapter

MCQ_SINGLE
Absolute minimum value of $f(x)=(x-2)^{2}+5$ in the interval [-3, 2] is :
VSA
Determine the values of x for which $f(x)=\frac{x-4}{x+1}$, $x\ne-1$ is an increasing or a decreasing function.
SUBJECTIVE
Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.
VSA
For the curve $y=5x-2x^{3}$ if x increases at the rate of $2\text{ units/s}$, then how fast is the slope of the curve changing when $x=2$?
LA
Find the absolute maximum and absolute minimum of function $f(x)=2x^{3}-15x^{2}+36x+1$ on $[1, 5]$.
View All Questions