# Understand the problem

Consider the following two statements:
• (E)Continuous functions on $latex[1 , 2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p(x) \in \Bbb R[x]$ such that $p(-x) = p(x))$.
• (O)Continuous functions on $[1 , 2]$ can be approximated uniformly by a sequence of odd polynomials (i.e., polynomials $p(x)\in \Bbb R[x]$ such that $p(-x) = -p(x))$.
Choose the correct option below.
1. (E) and (O) are both false
2. (E) and (O) are both true
3. (E) is true but (O) is false
4. (E) is false but (O) is true
##### Source of the problem
TIFR 2019 GS Part A, Problem 8
##### Topic
Weierstrass approximation theorem
Moderate

### Introduction to Real Analysis by Donald R. Sherbert and Robert G. Bartle

Do you really need a hint? Try it first!

$$g(x) = \begin{cases} f(x), & \text{if x \in [1,2] } \\[3ex] f(1), & \text{if x \in [-1,1] }\\[3ex] f(-x), & \text{if x \in [-2,-1] } \end{cases}$$. This is a continuous even function. Weierstrass’s approximation theorem, there is a $$g_n \to g(x)$$. Can you form an even sequence that converges?
The even seqn of polys are $[p_n(x)+p_n(x)]/2$. Prove that it converges to $g(x)$ on $[-2,2]$. Now think about restriction
For an odd function it is bit interesting. Consider a straight line $L(x)$ passing through $f(1)$ and $-f(1)$ observe that it passes through $(0,0)$. Consider the function $$h(x) = \begin{cases} f(x), & \text{if x \in [1,2] } \\[3ex] L(x), & \text{if x \in [-1,1] }\\[3ex] -f(-x), & \text{if x \in [-2,-1] } \end{cases}$$
There exists an odd function converging to $h(x)$[See the proof of Weierstrass’ theorem]. Restrict it.

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Similar Problems

## Sequences & Subsequences : IIT 2018 Problem 10

This problem appeared in IIT JAM 2018 whch pricisely reqiures concepts of sequences and subsequences from mathematical field real analysis

## Cyclic Groups & Subgroups : IIT 2018 Problem 1

This is an application abstract algebra question that appeared in IIT JAM 2018. The concept required is the cyclic groups , subgroups and proper subgroups.

## Acute angles between surfaces: IIT JAM 2018 Qn 6

This is an application analysis question that appeared in IIT JAM 2018. The concept required is the multivarible calculus and vector analysis.

## Finding Tangent plane: IIT JAM 2018 problem 5

What are we learning?Gradient is one of the key concepts of vector calculus. We will use this problem from IIT JAM 2018 will use these ideasUnderstand the problemThe tangent plane to the surface $latex z= \sqrt{x^2+3y^2}$ at (1,1,2) is given by $$x-3y+z=0$$...

## An excursion in Linear Algebra

Did you know Einstein badly needed linear algebra? We will begin from scratch in this open seminar and master useful tools on the way. The open seminar on linear algebra is coming up on 14th November 2019, (8 PM IST).

## Linear Algebra total recall (Open Seminar)

Open Seminar on linear algebra. A review of all major ideas. Even if you have little or no knowledge about Linear Algebra, you may join. Register now.

## 4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 2

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 1

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.