Vidaara.orgClass 12 · Mathematics
CodeVID-M12-06-INC-01
Increasing & Decreasing Functions — 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.
$f$ is strictly increasing on $I$ if, for all $x\in I$:
- A.$f'(x)<0$
- B.$f'(x)=0$
- C.$f'(x)>0$
- D.$f''(x)>0$
2.
$f(x)=x^2-4x+1$ is increasing on:
- A.$(-\infty,2)$
- B.$(2,\infty)$
- C.$\mathbb{R}$
- D.$(0,2)$
3.
$f(x)=x^3-3x$ is decreasing on:
- A.$(-1,1)$
- B.$(1,\infty)$
- C.$(-\infty,-1)$
- D.$\mathbb{R}$
4.
Which is increasing on all of $\mathbb{R}$?
- A.$x^2$
- B.$e^{x}$
- C.$\cos x$
- D.$-x$
5.
Critical points are where:
- A.$f=0$
- B.$f'(x)=0$ or undefined
- C.$f''=0$
- D.$x=0$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the interval where $f(x)=x^2-2x$ is increasing.
7.
Is $f(x)=e^x$ increasing or decreasing?
8.
Find the critical point of $f(x)=x^2-6x+5$.
9.
Is $f(x)=-x+5$ increasing or decreasing?
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the intervals of increase and decrease of $f(x)=x^2-4x+3$.
11.
Show that $f(x)=x^3+x$ is increasing on $\mathbb{R}$.
12.
Find where $f(x)=2x^3-3x^2-12x$ is increasing.
13.
Find where $f(x)=x^3-3x$ is decreasing.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Find the intervals in which $f(x)=x^4-2x^2$ is increasing and decreasing.
15.
Show that $f(x)=\sin x$ is increasing on $\left(0,\tfrac{\pi}{2}\right)$ and decreasing on $\left(\tfrac{\pi}{2},\pi\right)$.
Answer Key
Section A — Multiple Choice Questions
- (C) $f'(x)>0$
- (B) $(2,\infty)$
- (A) $(-1,1)$
- (B) $e^{x}$
- (B) $f'(x)=0$ or undefined
Section B — Short Answer (2 marks)
- $(1,\infty)$.
- Increasing.
- $x=3$.
- Decreasing.
Section C — Short Answer (3 marks)
- Decreasing on $(-\infty,2)$, increasing on $(2,\infty)$.
- $f'(x)=3x^2+1>0$, so increasing.
- $(-\infty,-1)\cup(2,\infty)$.
- $(-1,1)$.
Section D — Long Answer (5 marks)
- Increasing on $(-1,0)\cup(1,\infty)$; decreasing on $(-\infty,-1)\cup(0,1)$.
- Increasing on $\left(0,\tfrac{\pi}{2}\right)$, decreasing on $\left(\tfrac{\pi}{2},\pi\right)$.
Generated by Vidaara.org · Assignment VID-M12-06-INC-01 · vidaara.org