Question

If g(x) = f(x) + f(1 – x ) and f ¢¢(x) < 0, 0 ≤ x £ 1, then

• A. g is increasing in (0, 1/2)
• B. g is decreasing in (0, ¥)
• C. g is increasing in (– ¥, 1/2)
• D. g is decreasing in (1/2, ¥)

Answer 1

Answer: (A)

g is increasing in (0, 1/2)

Answer 2

f ¢¢(x) < 0 ̃ f ¢(x) is decreasing

g is increasing ̃ g ¢ (x) > 0

̃ f ¢(x) – f ¢(1 – x) > 0 ̃ f ¢(x) > f ¢ (1 – x)

̃ x < 1 – x ̃ x < $\frac{1}{2}$

\ g is increasing in x Î (0, 1/2)

g is decreasing when g ¢(x) < 0

f ¢(x) < f ¢ (1 – x)

x > 1 – x ̃ x > 1/2

\ g is decreasing x Î (1/2, 1)