If $a, b, c \in \mathbb{R}$ and satisfy $3a + 5b + 15c = 0$,then the equation $ax^4 + bx^2 + c = 0$ has:

Given that $f(x)$ is continuously differentiable on $a \le x \le b$ where $a < b, f(a) < 0$ and $f(b) > 0$,which of the following are always true?
$(i)$ $f(x)$ is bounded on $a \le x \le b$.
$(ii)$ The equation $f(x) = 0$ has at least one solution in $a < x < b$.
$(iii)$ The maximum and minimum values of $f(x)$ on $a \le x \le b$ occur at points where $f'(c) = 0$.
$(iv)$ There is at least one point $c$ with $a < c < b$ where $f'(c) > 0$.
$(v)$ There is at least one point $d$ with $a < d < b$ where $f'(d) < 0$.

If $f(x)$ is differentiable on the interval $[2, 5]$ such that $f(2) = 1/5$ and $f(5) = 1/2$,then there exists a number $c$ such that $2 < c < 5$ and $f'(c) = \dots$

If the function $f(x)=x^3+ax^2+bx+40$ satisfies the conditions of Rolle's theorem on the interval $[-5,4]$ and $-5,4$ are two roots of the equation $f(x)=0$, then one of the values of $c$ as stated in that theorem is

Let $f: D \rightarrow R$ where $D=[0,1] \cup [2,4]$ be defined by $f(x)=\begin{cases} x, & \text{if } x \in [0,1] \\ 4-x, & \text{if } x \in [2,4] \end{cases}$. Then,

If $f(x) = x^{\alpha} \log x, x > 0, f(0) = 0$ and $f(x)$ satisfies Rolle's theorem on $[0, 1]$,then what is the value of $\alpha$?