Q1. Are the following statements true or false? Give reasons for your answer.
- (a)) Complement of the open interval ]0,1] is an open set. (100 words)
- (b)) Every bounded sequence is not convergent. (100 words)
- (c)) The function f : [−2, 2] → R defined by f(x) = (4x + 3) / (x² + 1) is uniformly continuous. (100 words)
- (d)) If the first derivative of a function at a point vanishes, then it has an extreme value at that point. (100 words)
- (e)) The function f : [0, 2] → R defined by f(x) = x + [x] is not integrable. (100 words)
Key Points:
- A set's complement is not necessarily open, e.g., complement of ]0,1] includes boundary points.
- Bounded sequences are not always convergent; `(-1)^n` is a counterexample.
- Continuous functions on closed and bounded intervals (compact sets) are uniformly continuous.
- A vanishing first derivative (f'(c)=0) indicates a critical point, not necessarily an extremum (e.g., inflection point of x³ at x=0).
Answer: