Not a really wordy problem

Calculus Level 3

If the derivative of y = a x 3 + 9 x 2 + c x + d y = ax^3 + 9x^2 + cx + d has exactly one real root, where a a , b b , c c , and d d are nonzero reals, then determine the value of a c ac .


The answer is 27.

Efren Medallo by Efren Medallo , 26, Philippines

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1 solution

Efren Medallo
Jun 19, 2015

The derivative of y y is 3 a x 2 + 18 x + c 3ax^2 + 18x + c .

This will only have one real root if 1 8 2 − 4 ( 3 a ) ( c ) = 0 18^2 - 4(3a)(c) = 0

That is,

12 a c = 324 12ac = 324

so a c = 27 ac = 27 .

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