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If f(x) = x^(1//x) , " then: f''(e) is...

If ` f(x) = x^(1//x) , " then: f''(e)` is

A

1

B

e

C

`-e^(1//3)`

D

`- e^((1)/(e) - 3) `

Text Solution

Verified by Experts

The correct Answer is:
D
Given , ` f(x) = x^((1)/(x))`
` log f(x) = (1)/(x) log x `
` therefore (1)/(f(x)) xx f'(x) = (1)/(x) xx(1)/(x)+ log x (-(1)/(x^(2)))`
`rArr f'(x) = f(x) [(1 - log x)/(x^(2))]`
` rArr f''(x) = f(x) [ (x^(2) (-(1)/(x))- (1 - log x) 2x)/(x^(4))] + ((1 - log x)/(x^(2))) f'(x)`
Now , ` f(e) = e^(1//e)`
and ` f'(e) = 0 `
`therefore f''(e) = e^(1//e) [(-e -0)/(e^(4))] = - e^(1//e) . e^(-3)`
` =e-^(1//e-3)` .
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