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# If $\phi(x) = f(x) + f(1-x), f’(x)<0 \: for\: 0 \leq x \leq 1$, then the tautology among the following is

$\begin {array} {1 1} (A)\;\phi(x)\: is \: increasing \: in\: \bigg[0,\large\frac{1}{2} \bigg] & \quad (B)\;\phi(x)\: is\: decreasing \: in\: \bigg[0,\large\frac{1}{2} \bigg] \\ (C)\;\phi(x)\: is \: increasing \: in\: \bigg[ \large\frac{1}{2},1 \bigg] & \quad (D)\;\phi(x)\: has\: minima\: at\: x=\large\frac{1}{2} \end {array}$

Ans : (A)
So, $\phi’(x)=f’(x) – f’(1-x)$
For maxima or minima, put $\phi’(x)=0$
$f’(x)-f’(1-x)=0$
$\phi’\bigg(\large\frac{1}{2} \bigg)=0$
Now, $\phi’’(x)=f’’(x)+f’’(1-x)$
$\phi’’ \bigg( \large\frac{1}{2} \bigg)<0\: \: \: \: (f’’(x)<0)$
So, $x=\large\frac{1}{2}$ is a point of maxima.
Hence, function $\phi’(x)$ increases in $\bigg[0,\large\frac{1}{2} \bigg]$