Question

How many inflection points does the function have?

Answer 1

`f(x)=-\frac{x^2}2+x-\cos^2x`


`\implies~f'(x)=-x+1-2\cos x(-\sin x)=-x+\sin2x+1`


`\implies~f''(x)=-1+2\cos2x`

Inflection points always occur at points where the second derivative is equal to zero, but not every such point will be an actual inflection point.


`-1+2\cos2x=0\iff\frac12=\cos2x`

`\implies 2x=\pm\frac\pi3+2n\pi`

`\implies x=\pm\frac\pi6+n\pi`

where
`n` is an integer. There are infinitely many inflection points for this function.