Correct option is (c) -2
p(x) = 2x3 - 3ax2 - 12a2x - 2
p'(x) = 6x2 - 6ax - 12a2
p'(x) = 0 given
x2 - ax - 2a2 = 0
x = \(\frac{a\pm\sqrt{a^2+8a^2}}2\) = \(\frac{a\pm3a}2\) = 2a or -a
p"(x) = 12x - 6a
p"(-a) = -12a - 6a = -18a
p"(2a) = 24a - 6a = 18a
p(\(\frac{s+t}2\)) = p(\(\frac{2a+(-a)}2\)) = p(\(\frac a2\)) = -54 (Given)
⇒ \(\frac{a^3}4-\frac{3a^3}4-\frac{12a^3}2-2=-54\)
⇒ -2a3 - 24a3 = -208
⇒ a3 = -208/-26 = 8 = 23
⇒ a = 2
∴ f"(-a) = -18 x 2 < 0
f"(2a) = 18 x 2 > 0
∴ x = -a is point of maxima
x = 2a is point of minima
∴ s = -a = -2