# What are the points of inflection of #f(x)=x^3sin^2x # on the interval #x in [0,2pi]#?

##### 1 Answer

The successive graphs reveal POI at x = 0, +-1.24, +-2.55, +-3.68, +-4.82, ....

#### Explanation:

Excepting x = 0, the other zeros of f'', in infinitude, cannot be

obtained in mathematical exactitude.

Yet, we can locate them in pairs in between

There are 4 POI, for

They are nearly 1.24, 2.55, 3.58 and 4.82, nearly..

Some locations near O have been obtained by graphical method,

using appropriate sub domains and ranges.

graph{x^3sinxsinx [-20, 20, -10, 10]}

Graph 1 : Graph is symmetrical about O.

graph{x^3sinxsinx [0, 6.28, -50, 300]}

Graph 2 : A revelation of four POI and the power-growth of local maxima of y

graph{x^3sinxsinx [1.24 1.26, -10, 10]}

Graph 3 : locates a POI, near 1.24.

graph{x^3sinxsinx [2.54, 2.56,-10, 10]}

Graph 4 : Locates a POII, near 2.55.

graph{x^3sinxsinx [3.56 3.59,-10, 10]}

graph{x^3sinxsinx [4.81 4.83, 0, 200]}