First Derivative Test for Local Minimum

Q. Let $g\left(x\right)=2f\left(\frac{x}{2}\right)+f(2-x)$ and $f^{\prime \prime }\left(x\right)<0,\forall x\in (0,2).$ Then which of the following is (are) TRUE?

1. $g\left(x\right)$ is decreasing in $(0,\frac{4}{3})$
2. $g\left(x\right)$ is decreasing in $(\frac{4}{3},2)$
3. $g\left(\frac{1}{10}\right)<g\left(\frac{11}{10}\right)$
4. $g\left(x\right)$ has a local minimum at $x=\frac{4}{3}$

Q.

Match the entries of col. I with those of col. II.
$\begin{array}{cc}Column-I& Column-II\\ \left(a\right)f\left(x\right)=\frac{1-x+x^2}{1+x-x^2}\text{on}[0,1]& \left(p\right)Greatestvalueoff=1\\ \left(b\right)f\left(x\right)=2tanx-ta{n}^{2}x\text{on}[0,\frac{\pi }{2}]& \left(q\right)Leastvalueoff=\frac{3}{5}\\ \left(c\right)f\left(x\right)=\frac{2}{\pi }(sin2x-x)\text{on}[-\frac{\pi }{2},\frac{\pi }{2}]& \left(r\right)Leastvalueoff=-1\\ \left(d\right)f\left(x\right)=\frac{1}{2},(x^3-3x^2+6x-2)\text{on}(-1,1)& \left(s\right)Leastvalueoff=-6\end{array}$

1. $\left(a\right)\to \left(p\right),\left(b\right)\to \left(q\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(s\right)$

2. $\left(a\right)\to (p,q),\left(b\right)\to \left(p\right),\left(c\right)\to (p,r),\left(d\right)\to (p,s)$

3. $\left(a\right)\to \left(q\right),\left(b\right)\to \left(p\right),\left(c\right)\to \left(p\right),\left(d\right)\to \left(s\right)$

4. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(q\right),\left(c\right)\to (p,r),\left(d\right)\to (r,s)$