Let f(x) be a cubic polynomial with real coefficients. Suppose that 

f(x) has exactly one real root and that root is simple.

Which one of the following statements holds for all antiderivatives F(x) of f(x) ?

​​​​

$$\lim_{n \to \infty} \frac{((n+1)(n+2) \cdots (n+n))^{1/n}}{n}.$$​

Suppose $$f:[0,1]\times[0,1] \to (0,1)\times (0,1)$$ is a continuous,

non constant function. Which of the following statements 

are NOT true ?

Let f be a non constant polynomial of degree k and let g:$$R\to R$$ be a bounded continuous function

Which of the following statements are necessarily true? 

​Let S be a dense subset of R and,

Which of the following statements is true ?

​$$\text{Let } (X, d) \text{ be a metric space and let } f: X \to X \text{ be a function such that:}\\d(f(x), f(y)) \leq d(x, y), \quad \text{for every } x, y \in X.\\\text{Which of the following statements is necessarily true?}$$​​

Let X be a non empty finite set and,

​$$\text{Given } f, g \text{ are continuous functions on } [0, 1] \text{ such that } \\f(0) = f(1) = 0, \; g(0) = g(1) = 1, \; \text{and } f(1/2) > g(1/2). \\\text{Which of the following statements is true?}$$​​

​​

$$\lim_{n \to \infty} \frac{((n+1)(n+2) \cdots (n+n))^{1/n}}{n}.$$​