Test for Monotonicity about a Point

Q. Let $f\left(x\right)=\{\begin{array}{cc}x+2,& -1\le x<0\\ 1,& x=0\\ \frac{x}{2},& 0<x\le 1\end{array}$
Then in $[-1,1],$ this function has

1. a minimum
2. a maximum
3. either a maximum or a minimum
4. neither a maximum nor a minimum

Q. $f\left(x\right)$ is cubic polynomial with $f\left(2\right)=18$ and $f\left(1\right)=-1.$ Also $f\left(x\right)$ has local maxima at $x=-1$ and $f\text{'}\left(x\right)$ has local minima at $x=0$, then

1. the distance between $(-1,2)$ and $(a,f(a\left)\right)$, where $x=a$ is the point of local minima is $2\sqrt{5}$
2. $f\left(x\right)$ is increasing for $x\in [1,2\sqrt{5}]$
3. $f\left(x\right)$ has local minima at $x=1$
4. the value of $f\left(0\right)=15$

Q.

f(x) = sinx is decreasing about $x=\pi$.

1. True

2. False

Q.

Find the point(s) where the function f(x) = $x^3$ - 3x + 2 is increasing

1. x = 0

2. x = 1

3. x = 2

4. None of these

Q. If $2\le x\le 4,$ then the maximum value of
$f\left(x\right)=(x-2{)}^6(4-x{)}^5$ is

1. ${\left(\frac{2}{11}\right)}^6{\left(\frac{10}{11}\right)}^5$
2. ${\left(\frac{12}{11}\right)}^6{\left(\frac{10}{11}\right)}^5$
3. ${\left(\frac{11}{12}\right)}^6{\left(\frac{11}{10}\right)}^5$
4. ${\left(\frac{2}{11}\right)}^6{\left(\frac{10}{9}\right)}^5$