if-g-x-f-x-f-1-x-and-f-2-x-lt-0-then-show-that-g-x-is-increasing-in-0-1-2-and-g-x-is-decreasing-in-1-2-1-

Question Number 27204 by rish@bh last updated on 03/Jan/18

if g (x) = f (x) + f (1 - x)

and f^{(2)} (x) < 0

then show that

g (x) is increasing in (0, 1 / 2) and

g (x) is decreasing in (1 / 2, 1)

Answered by Giannibo last updated on 03/Jan/18

\exists g^{'} because \exists f^{″} \Rightarrow \exists f^{'}

g^{'} (x) = f^{'} (x) - f^{'} (1 - x)

f^{″} (x) < 0 \Rightarrow f^{'} ↓

If x < 1 - x \Leftrightarrow x < \frac{1}{2}

\overset{f^{'} ↓}{\Rightarrow} f^{'} (x) > f^{'} (1 - x)

g^{'} (x) > 0 \Rightarrow g ↑

If x > 1 - x \Leftrightarrow x > \frac{1}{2}

\overset{f^{'} ↓}{\Rightarrow} f^{'} (x) < f^{'} (1 - x)

g^{'} (x) < 0 \Rightarrow g ↓

Commented by rish@bh last updated on 03/Jan/18

Thank you!

Commented by Penguin last updated on 04/Jan/18

What does f^{'} ↓ mean ? ?

Commented by Giannibo last updated on 04/Jan/18

f^{'} is decreasing