Application of DerivativeHard
Question
Let f(x) = (x2 - 1)n (x2 + x + 1). f(x) has local extremum at x = 1 if
Options
A.n = 2
B.n = 3
C.n = 4
D.n = 6
Solution
f′(x) = (x - 1)n - 1 (x + 1)n - 1 [2(n + 1)x3 + (2n + 1)x2 + 2(n - 1)x - 1]
At x = 1 2(n + 1)x3 + (2n + 1)x2 + 2 (n - 1)x - 1 ≠ 0
for n ∈ N
∴ n - 1 must be odd
⇒ n is even
At x = 1 2(n + 1)x3 + (2n + 1)x2 + 2 (n - 1)x - 1 ≠ 0
for n ∈ N
∴ n - 1 must be odd
⇒ n is even
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