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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