Question

Let f(x) = (x – a)ⁿg(x) , where g⁽ⁿ⁾(a) ≠ 0; n = 0, 1, 2, 3.... then ______.

Options

• f(x) has local extremum at x = a, when n = 3

• f(x) has local extremum at x = a; when n = 4

• f(x) has neither local maximum nor local minimum at x = a, when n = 2

• f(x) has neither local maximum nor local minimum at x = a, when n = 4

Answer 1

Let f(x) = (x – a)ⁿg(x) , where g⁽ⁿ⁾(a) ≠ 0; n = 0, 1, 2, 3.... then f(x) has local extremum at x = a; when n = 4.

Explanation:

f'(x) = n(x – a)^n–1g(x) + g'(x)(x – a)ⁿ

= `(x - a)^n[(ng(x))/((x - a)) + g^'(x)]`

= (x – a)^n–1[ng(x) + g'(x)(x – a)]

For extremum at x = a

(n – 1) must be odd

n is even