Let g(x) = 2f (x/2) + f(2 –x) and f ′′ (x) < 0 for all x ∈ (0, 2). Find the intervals of increase and decrease of g(x).
Text Solution
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Sol. We have,
f ′′ (x) < 0
⇒ f ′ (x) is decreasing on (0, 2)
⇒ f ′ 
> f ′ (2 –x), if 
< 2 – x and,
f ′ 
< f ′ (2 –x) , if 
> 2 – x
⇒ f ′ 
> f ′ (2 – x), if x < 
and,
f ′ 
> f ′ (2 – x), if x > 
⇒ f ′ 
> f ′ (2 –x) , if x ∈ (0, 4/3) and
f ′ 
< f ′ (2 –x) if x ∈ (4/3, 2)
⇒ f ′ 
– f ′ (2 –x ) > 0,if x ∈ (0, 4/3) and,
f ′ 
– f ′ (2 –x) < 0 if x ∈ (4/3, 2) ... (i)
Now,
g(x) = 2f 
+ f(2 –x),
⇒ g ′ (x) = f ′ 
– f ′ (2 –x),
⇒ g ′ (x) > 0 if x ∈ (0, 4/3) [Using (i)]
and,
g ′ (x) < 0 if x ∈ (4/3, 2)
⇒ g(x) is increasing on (0, 4/3) and decreasing on (4/3, 2)
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