Which of the following function/functions has/have point of inflection? `f(x)=x^(6/7)` (b) `f(x)=x^6` `f(x)=cosx+2x` (d) `f(x)=x|x|`

Options

A. f(x)=`x^(6//7)`
B. `f(x)=x^(6)`
C. f(x)=cosx+2x
D. f(X)=x|x|

Answer

3,4

Explanation

`f(x)=x^(6//7)`
`f(X)=-(6)/(7)(13)/(7)x^(-8/7)`
here f(X) does not changes sign hence it has no point of inflection

`For f(X) =x^(6)f(x)=30x^(4)` but f(x) does not changes sign in the neighborhood of x =0
f(x)=cosx +2x
or f(x)==cos x
or `f(x) =0 for x=(2n+1)pi//2n in Z`
Also sing of f(x) changes sign in the neightborhood of `(2n+1)pi//2` hence fucntion has infintie points of inflection
`f(x)=x|x|={{:(-x^(2),xlt0),(x^(2),xge0):}`
or `f(x)={{:(-2,xlt0),(2,xgt0):}`
Here f(x) changes sign in the neightborhood of x=0 hence has point of inflection