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.
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.
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