Let (x)=f(x)+f(2a-x) and f'(x) > 0 for all x [0,a]. Then, (x): - Vidyadip

MCQ

Let $\phi(\text{x})=\text{f}(\text{x})+\text{f}(2\text{a}-\text{x})$ and f'(x) > 0 for all $\text{x}\in[0,\text{a}].$ Then, $\phi(\text{x}):$

A
Increases on [0, a]
✓
Decreases on [0, a]
C
Increases on [-a, 0]
D
Decreases on [a, 2a]

✓

Answer

Correct option: B.

Decreases on [0, a]

$\phi(\text{x})=\text{f}(\text{x})+\text{f}(2\text{a}-\text{x})$
$\phi'(\text{x})=\text{f}'(\text{x})-\text{f}'(2\text{a}-\text{x})$
$\text{f}''(\text{x})>0$ as $\text{f}'(\text{x})>0$
Considering $\text{x}\in[0,\text{a}]$
$\text{x}\leq2\text{a}-\text{x}$
$\text{f}'(\text{x})\leq\text{f}(2\text{a}-\text{x})$
Also, $\phi(\text{x})=\text{f}'(\text{x})-\text{f}'(2\text{a}-\text{x})$
$\phi(\text{x})$ is decreasing on [0, a]

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