Answer to Question #128970 in Real Analysis for Neha Rasaily

Question

the following statements true or false? Give reasons for your answer.

 a) For the function f, defined by f(x) =4x²-4x²- 7x -2,there exists a point 

C ∈ ]-1/2,2[ satisfying f′(c) = 0

 b) For all even integral values of n,

lim (x+1)^-n exists.

x→∞

 c) The function f defined by f(x)= [x − 1], (where [x] is the greatest integer 

function) is integrable on the interval [2,-4].

 d) Every infinite set is an open set. 

 e) All strictly monotonically decreasing sequences are convergent.

Answer 1

a). $f'(c)=-7$ for all $c\in{\mathbb{R}}$ . Thus, the statement is false.

b). We assume that $n>0$ . For $n=2,4,...$ we have

$$\lim_{x\rightarrow\infty}\frac{1}{(1+x)^n}=0.$$

For $n=-2,-4,...$ $\lim_{x\rightarrow\infty}\frac{1}{(1+x)^n}=+\infty.$

c). The function is integrable, since it has a finite number of discontinuity points. Namely, $f(x)$ has discontinuities at n=-3,-2,-1,0,1.

d). The statement is wrong. E.g., we can take integer numbers and a ball of an arbitrary radius with a center at the point 1. It will never belong to the set of integers irrespectively of the radius.

e). We can take the sequence $-n$ , where n=1,2,... It decreases and is not convergent.