Question

If $\phi(x)=f(x)+f(2 a-x)$ and $f^{\prime \prime}(x)>0, a>0,0 \leq x \leq 2 a$ then

• A. $\phi( x )$ increases in $( a , 2 a )$
• B. $\phi(x)$ increases in $(0, a)$
• C. $ \phi( x )$ decreases in $(0, a )$
• D. $\phi(x)$ decreases in $(a, 2 a)$

Answer 1

Answer: (A), (C)

$ \phi(x)=f(x)+f(2 a-x)$
$\phi^{\prime}(x)=f^{\prime}(x)+f^{\prime}(2 a-x) $
$\text { Since } f^{\prime \prime}(x)>0 \Rightarrow f^{\prime}(x) \text { is an increasing function } $
$\text { hence } f^{\prime}(x)>f \phi(2 a-x) \Rightarrow x>2 a-x \Rightarrow x>a$
$\therefore f^{\prime}(x)>0$ if $x>a \Rightarrow f(x)$ is $\uparrow$ in $(a, 2 a)$
$ \|\| ^{\text {ly }} f ( x )$ is $\downarrow$ in $(0, a ) $