The interval in which the function $f(x) = \frac{\log(7+x)}{\log(3+x)}$ for $x > 0$ decreases is:

The function $f(x)=3x^{4}+16x^{3}-30x^{2}+10$ is increasing for

Let $\phi(x) = f(x) + f(2a - x)$, $x \in [0, 2a]$ and $f^{\prime \prime}(x) > 0$ for all $x \in [0, a]$. Then $\phi(x)$ is

Which statement among the following is true?
$(i)$ The function $f(x) = x|x|$ is strictly increasing on $R - \{0\}$.
$(ii)$ The function $f(x) = \log_{(1/4)} x$ is strictly increasing on $(0, \infty)$.
$(iii)$ $A$ one-one function is always an increasing function.
$(iv)$ $f(x) = x^{1/3}$ is strictly decreasing on $R$.

If $y = ax^3 + 3x^2 + (2a + 1)x + 1000$ is a strictly increasing function for all values of $x$,then:

Let $h(x) = f(x) - \{f(x)\}^2 + \{f(x)\}^3$ for every real number $x$,then