If `f(x)={{:(x^(3)(1-x)",",xle0),(xlog_(e)x+3x",",xgt0):}` then which of the following is not true?

Options

A. f(x) has point of maxima at x = 0
B. f(x) has point minima at `x=e^(-4)`
C. f(x) has range R
D. none of these

Answer

D

Explanation

`f(x)={{:(x^(3)(1-x)",",xle0),(x ln x+3x",",xgt0):}`
`f(0)=0 ("using "x^(3)(1-x))`
`f(0^(-))=0`
`f(0^(+))=underset(xrarr0)(lim)(x ln x+3x)`
`=underset(xrarr0^(+))(lim)(lnx)/((1)/(x))+0`
`=underset(xrarr0^(+))(lim)((1)/(x))/(-(1)/(x^(2)))`
= 0
Thus, function is continuous at x = 0.
`f(0)gtf(0^(-)),f(0)gtf(0^(+))`
`therefore x= 0` is point of maxima.
Also `(x ln x+3x)'=4+lnx`
`4+lnx=0 rArr x=e^(-4)`
`(x ln x+3x)''=1//x gt 0 "for "xgt0`
Thus, `x=e^(-4)` is point of minima
`underset(xrarr-oo)(lim)x^(3)(1-x)=-oo`
`"and "underset(xrarroo)(lim)(xlnx+3x)=oo`
`therefore" Range of f(x) is R."`