Let $f: R \to R$ be a function. Define $g: R \to R$ by $g(x) = |f(x)|$ for all $x$. Then $g$ is

If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$,$x \in R$,then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to :

The function $f(x) = [|x|] - |[x]|$ where $[x]$ denotes the greatest integer function:

If $f(x) = x \left( \frac{1}{x-1} + \frac{1}{x} + \frac{1}{x+1} \right)$ for $x > 1$, then:

$[t]$ denotes the greatest integer function and $[t-m] = [t] - m$ when $m \in \mathbb{Z}$. If $k = 2[2x - 1] - 1$ and $3[2x - 2] + 1 = 2[2x - 1] - 1$,then the range of $f(x) = [k + 5x]$ is

Consider the function $f: [\frac{1}{2}, 1] \rightarrow \mathbb{R}$ defined by $f(x) = 4\sqrt{2}x^3 - 3\sqrt{2}x - 1$. Consider the following statements:
$(I)$ The curve $y = f(x)$ intersects the $x$-axis exactly at one point.
$(II)$ The curve $y = f(x)$ intersects the $x$-axis at $x = \cos \frac{\pi}{12}$.
Then: