Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by

$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \neq 0, \\ \dfrac{7}{3}, & \text{if } x = 0. \end{cases}$

Then which of the following statements is (are) TRUE?

Let

$$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$$ g(x)=2^{\alpha x}+2^{\alpha(1-x)} . $$

Then, which of the following statements is/are TRUE ?

Let f : R $$ \to $$ R be given by

$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define

$$F(x) = \int\limits_0^x {f(t)dt} $$, x > 0

Then which of the following options is/are correct?

Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0

Let x₁ < x₂ < x₃ < ... < x_n < ... be all the points of local maximum of f and y₁ < y₂ < y₃ < ... < y_n < ... be all the points of local minimum of f.

Then which of the following options is/are correct?