If the function f(x) = 2x3 – 9ax2 + 12a2x + 1,a > 0 has a local maximum at x = α and a local minimum at x = α2 then α and α2 are the roots of the equation:

Engineering

Mathematics

Maxima and Minima

Question

If the function f(x) = 2x³– 9ax²+ 12a²x + 1,a > 0 has a local maximum at x = α and a local minimum at x = α²then α and α² are the roots of the equation:

x²– 6x + 8 = 0

8x² + 6x – 1 = 0

x² + 16x + 8 = 0

8x² – 6x + 1 = 0

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f(x) = 2x³ – 9ax² + 12a²x + 1 ........ a > 0
f'(x) = 6x²– 18ax + 12a²
= 6[x² – 3ax + 2a²]
= 6 (x – a) (x – 2a)

Maxima at x = a = α
minima at x = 2a = α²
2a = a²
a = 2
α = 2α² = 4
x² – 6x + 8 = 0