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

Jan 28, 2017

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

#### Explanation:

$f = \frac{1}{2} {x}^{3} \left(1 - \cos 2 x\right)$, giving x-intercepts $0 , 0 , \pm \pi , \pm 2 \pi , \pm 3 \pi , \ldots$

$f ' ' = \frac{1}{2} \left(\left({x}^{3}\right) ' ' - \left({x}^{3} \cos 2 x\right) ' '\right)$

$= x \left(3 - \left(3 \cos 2 x - 6 x \sin 2 x - 2 {x}^{2} \cos 2 x\right)\right)$

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

$\left(k \pi . \left(k + 1\right) \pi\right) \mathmr{and} \left(- \left(k + 1\right) \pi , - k \pi\right) , k = 1 , 2 , 3 , - - -$, respectively.

There are 4 POI, for $x \in \left(0. 2 \pi\right)$.

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]}