Let f: [ 0,2 ] R be a twice differentiable function such that f''… - Vidyadip

MCQ

Let $f:\left[ {0,2} \right] \to R$ be a twice differentiable function such that $f''\left( x \right) > 0$, for all $x \in \left( {0,2} \right)$. If $\phi \left( x \right) = f\left( x \right) + f\left( {2 - x} \right)$, then $\phi $ is

A
increasing on $(0, 2)$
B
decreasing on $(0, 2)$
✓
decreasing on $(0, 1)$ and increasing on $(1, 2)$
D
increasing on $(0, 1)$ and decreasing on $(1, 2)$

✓

Answer

Correct option: C.

decreasing on $(0, 1)$ and increasing on $(1, 2)$

c
$\phi(x)=f(x)+f(2-x)$

$\phi^{\prime}(x)=f(x)-f^{\prime}(2-x)$ ........$(1)$

since $f^{\prime \prime}(x)>0$

$\Rightarrow f(x)$ is increasing $\forall x \in(0,2)$

Case $-1:$ When $x>2-x \Rightarrow x>1$

$\Rightarrow \phi^{\prime}(x)>0 \forall x \in(1,2)$

$\therefore $ $\cdot \phi(x)$ is increasing on $(1,2)$

Case $-11:$ When $x<2-x \Rightarrow x<1$

$\Rightarrow \phi^{\prime}(x)<0 \forall x \in(0,1)$

$\therefore $ $\phi(x)$ is decreasing on $(0,1)$

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