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in Integrals calculus by (476 points)
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A polynomial function p(x) = 2x- 3ax- 12a2x - 2 attains its local maxima at x = s and local minima at x = t. If \(p\frac{(s+t)}{2}\) = -54, then the value of 's' is

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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

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