If the function $f(x) = \frac{t + 3x - x^2}{x - 4}$,where $t$ is a parameter,has a local maximum and a local minimum,then the range of values of $t$ is:

If $f(x) = 7e^{\sin^2 x} - e^{\cos^2 x} + 2$,then $\sqrt{7f_{\min} + f_{\max}}$ is equal to

For all real $x$,the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $(x \ln x) f'(x) + (\ln x + 1) f(x) \geq 1$ for all $x \in [2,4]$,with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:
$(A): f(x) \leq 1$ for all $x \in [2,4]$
$(B): f(x) \geq \frac{1}{8}$ for all $x \in [2,4]$
Then,

If $x = p$ and $x = q$ are the points of local maxima and local minima respectively for the function $f(x) = x^5 - 5x^4 + 5x^3 - 10$,then:

If $y = \frac{ax - b}{(x - 1)(x - 4)}$ has a turning point $P(2, -1)$,then the values of $a$ and $b$ are