CBSE Class 12 Mathematics Applications of Derivatives Q #1699
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
Absolute minimum value of $f(x)=(x-2)^{2}+5$ in the interval [-3, 2] is :
(A) -3
(B) 2
(C) 5
(D) 30
Prev Next
Correct Answer: C

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Identify the function and interval

The given function is $f(x) = (x-2)^2 + 5$ defined on the closed interval $[-3, 2]$.

Step 2: Find the critical points

To find the critical points, we calculate the derivative $f'(x)$ and set it to zero: $$f'(x) = 2(x-2) = 0$$ This gives $x = 2$. Since $x = 2$ is an endpoint of the interval, we evaluate the function at the critical point and the boundaries.

Step 3: Evaluate the function at boundaries

We evaluate $f(x)$ at $x = -3$ and $x = 2$: $$f(-3) = (-3-2)^2 + 5 = (-5)^2 + 5 = 25 + 5 = 30$$ $$f(2) = (2-2)^2 + 5 = 0 + 5 = 5$$

Step 4: Determine the absolute minimum

Comparing the values $f(-3) = 30$ and $f(2) = 5$, the absolute minimum value is $5$.

Final Answer: 5

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the Extreme Value Theorem to find the absolute extrema of a function on a closed interval.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a specific sequence of steps: differentiating, identifying critical points, and evaluating endpoints.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to handle optimization problems, which is a core component of the Application of Derivatives unit.
||KEY:C

More from this Chapter

VSA
Show that the function $f(x)=4x^{3}-18x^{2}+27x-7$ has neither maxima nor minima.
MCQ_SINGLE
Let \(f(x)=|x|\), \(x\in R\). Then, which of the following statements is **incorrect**?
VSA
If M and m denote the local maximum and local minimum values of the function $f(x)=x+\frac{1}{x}(x\ne0)$ respectively, find the value of $(M-m)$
SUBJECTIVE
Water is essential for life, and rivers are vital water sources across the world. The Yamuna Action Plan (YAP), one of India's largest river restoration projects, was launched to clean the Yamuna, but its effectiveness (WQI) the Water Quality Index must be measured through cost-benefit, economic and environmental analysis. Therefore, comprehensive pollutant treatment, prevention of further contamination, and community awareness are crucial for sustainable river management. Implementing the same concept to the river Yamuna, if an initial base line WQI is 35, a quadratic expression is formed to measure the impact of investment of ₹$x$ crore on the WQI i.e., $W(x)$ as $W(x) = 35 + 1.6x - 0.02x^2$, where $x$ is in Rupees in crores Use the given $W(x)$ to answer the following questions: (i) Find the value of $x$ at the stationary point for the function $W(x)$. [1 mark] (ii) Find the maximum value of the WQI using the second derivative test? [1 mark] (iii) (a) If the investment is ₹ 35 crores, find by how much the WQI is short of its maximum achievable value.[2 marks] Or (b) Evaluate the WQI when the investment is ₹60 crore. Determine whether this indicates an enhancement or a diminishing return in our objective when compared with the optimum WQI. [2 marks]
VSA
Find the maximum and minimum values of the function given by \(f(x)=5+\sin 2x\).
View All Questions