# What are the extrema of f(x)=-sin^2(ln(x^2))-cos^2(ln(x^2)) on the interval [0,2pi]?

$f \left(x\right) = - \left[{\sin}^{2} \left(\ln \left({x}^{2}\right)\right) + {\cos}^{2} \left(\ln \left({x}^{2}\right)\right)\right]$
Recall that ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$:
$f \left(x\right) = - 1$
$f$ is a constant function. It has no relative extrema and is $- 1$ for all values of $x$ between $0$ and $2 \pi$.