# What are the points of inflection, if any, of f(x) = 5xcos^2x − 10sinx^2  on x in [0,2pi]?

Jan 2, 2016

Find $f ' ' \left(x\right)$ and determine where it changes signs.

$f ' \left(x\right) = 5 {\cos}^{2} x - 10 x \cos x \sin x - 20 x \cos \left({x}^{2}\right)$

$f ' ' \left(x\right) = - 20 \cos x \sin x + 10 x {\sin}^{2} x - 10 x {\cos}^{2} x - 20 \cos \left({x}^{2}\right) + 40 {x}^{2} \sin \left({x}^{2}\right)$

Graph of $f ' ' \left(x\right)$:

graph{-20cos(x)sin(x)+10x(sinx)^2-10x(cosx)^2-20cos(x^2)+40x^2sin(x^2) [-0, 6.283, -1800, 1800]}

There are a slew of times when $f ' ' \left(x\right)$ switches from positive to negative on the interval, all of which are points of inflection.

They are:

• $x = 0.886$
• $x = 1.854$
• $x = 2.521$
• $x = 3.066$
• $x = 3.559$
• $x = 3.996$
• $x = 4.341$
• $x = 4.698$
• $x = 5.011$
• $x = 5.321$
• $x = 5.607$
• $x = 5.878$
• $x = 6.144$

Graph of $f \left(x\right)$:

graph{5x(cosx)^2-10sin(x^2) [0, 6.283, -15, 40]}