Vidaara.orgClass 12 · Mathematics
CodeVID-M12-06-MAX-01
Maxima & Minima — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
At a critical point with $f''(c)>0$, $f$ has a local:
- A.maximum
- B.minimum
- C.inflection
- D.none
2.
$f(x)=x^2-6x+5$ has a minimum at:
- A.$x=0$
- B.$x=3$
- C.$x=5$
- D.$x=-3$
3.
Two parts of $20$ with maximum product are:
- A.$5,15$
- B.$10,10$
- C.$1,19$
- D.$8,12$
4.
Absolute extrema on $[a,b]$ are found among:
- A.only critical points
- B.only endpoints
- C.critical points and endpoints
- D.inflection points
5.
At a local maximum, $f'(c)=$
- A.$1$
- B.$0$
- C.$\infty$
- D.undefined
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the critical point of $f(x)=x^2-4x$.
7.
Does $f(x)=x^3$ have a local extremum at $x=0$?
8.
Find the maximum value of $f(x)=-(x-2)^2+5$.
9.
Two positive numbers have sum $10$. When is their product maximum?
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the local maximum and minimum of $f(x)=x^3-3x$.
11.
Find the absolute maximum of $f(x)=x^3-3x$ on $[0,2]$.
12.
Find two numbers whose sum is $24$ and whose product is maximum.
13.
Find the minimum value of $f(x)=x^2-6x+5$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
A square sheet of side $18$ cm has equal squares of side $x$ cut from each corner and the flaps folded to make an open box. Find $x$ for maximum volume and the maximum volume.
15.
A wire of length $28$ cm is cut into two pieces; one is bent into a square and the other into a circle. Find the lengths so that the total area is minimum.
Answer Key
Section A — Multiple Choice Questions
- (B) minimum
- (B) $x=3$
- (B) $10,10$
- (C) critical points and endpoints
- (B) $0$
Section B — Short Answer (2 marks)
- $x=2$.
- No (point of inflection).
- $5$.
- When both are $5$.
Section C — Short Answer (3 marks)
- Max $2$ at $x=-1$; min $-2$ at $x=1$.
- $2$ (at $x=2$).
- $12$ and $12$.
- $-4$ (at $x=3$).
Section D — Long Answer (5 marks)
- $x=3$ cm; maximum volume $=432$ cm$^3$.
- Square piece $=\dfrac{112}{\pi+4}$ cm, circle piece $=\dfrac{28\pi}{\pi+4}$ cm.
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