(B) We are given that (1) f ′(a) > 0; (2)f ′′(a) < 0; and (3) G ′(a) < 0. Since G ′(x) = 2 f (x) · f
′(x), therefore G ′(a) = 2f (a) · f (a). Conditions (1) and (3) imply that (4)f (a) < 0. Since G
′′(x) = 2[f (x) · f ′′ (x) + (f ′(x))^2 ], therefore G′′(a) = 2[f (a)f ′′ (a) + (f′ (a))^2 ]. Then the sign
of G ′′(a) is 2[(−) · (−) + (+)] or positive, where the minus signs in the parentheses follow
from conditions (4) and (2).
