The number of turning points of the curve $f(x) = 2 \cos x - \sin 2x$ in the interval $[-\pi, \pi]$ is

For a positive real number $x$,when its reciprocal is added to it,the sum attains a minimum value. What is the value of $x$?

If a line is moving between the coordinate axes such that the sum of the intercepts made by it on the coordinate axes is always $12$,then the equation of that line which forms a triangle of maximum area with the coordinate axes is

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

If $ax + \frac{b}{x} \ge c$ for all positive $x$,where $a, b > 0$,then:

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?