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

With the exception of x = 0, all of f''s zeros cannot be

attained with exact mathematical precision.

However, we are able to find them in pairs between

1.24, 2.55, 3.58, and 4.82 are almost exactly the same.

A graphical method has been used to obtain some locations near O.

utilizing the proper ranges and subdomains.

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

Graph 1: The graph exhibits symmetry around O.

graph{x^3sinxsinx [0, 6.28, -50, 300]} Graph 2: Four points of interest and the power-growth of y's local maxima are revealed

graph{x^3sinxsinx [1.24 1.26, -10, 10]} Figure 3: finds a point of interest close to 1.24.

A POII is located in the vicinity of 2.55, as shown in graph{x^3sinxsinx [2.54, 2.56,-10, 10]}.

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

To find the points of inflection, we need to find where the second derivative changes sign. The second derivative of $f(x) = x^3 \sin^2(x)$ is:

$f''(x) = 6x \sin^2(x) + 2x^3 \sin(x) \cos(x)$

The points of inflection occur where $f''(x) = 0$ or is undefined, and where $f''(x)$ changes sign. Solving $f''(x) = 0$ and analyzing the sign changes on the interval $[0, 2\pi]$ will give us the points of inflection.