Draw the graph of `f(x) = 2x+3x^(2//3)` and discuss the type of non-differentiability for the function. Also find the point of inflection.

Explanation

We have `f(x) = 2x+3x^(2//3)`
`therefore f^(')(x)=2+3. 2/3x^(-1/3)=2(1+x^(-1//3))`
`therefore f^(')(x) = 2+3. 2/3x^(-1//3)=2(1+x^(-1//3))` ltbr Let `f^(')(x)=0`
`rArr x^(1//3)+1 =0 rArr x=-1`
`rArr f^('')(x) = -2/3x^(-1)`
and `f^('')(-1) =-2/3(-1)^(4//3) = -2/3 lt 0`
So `x=-1` is the point of maxima.
Also, `f^(')(0)` does not exist due to the vertical tangent.
Hence `f(x)` is non-differentiable at `x=0`
The graph of the function is as shown in the following figure.

From the graph, `x=0` is the point of local minima.