Let g(x) be an odd continuous function from R → R such that g(n) = (–1)n · (n + 1), if n is a prime natural number then find the minimum number of real roots of g(x) = 0, \forall x \in R.

Engineering

Mathematics

Intermediate Value Theorem

Question

Let g(x) be an odd continuous function from R → R such that g(n) = (–1)ⁿ · (n + 1), if n is a prime natural number then find the minimum number of real roots of g(x) = 0, $\forall$ x $\in$ R.

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$∵$ g(–x) = –g(x) ⇒ g(0) = 0
g(2) = 3 $∴$ g(–2) = –3
g(3) = –4 g(3) = 4
$∴$ Minimum number of roots of g(x) = 0 will be 3.