APFIVE A researcher studies the behavior of the polynomial function
$$f(x)=-x^4+4*x^2$$. After computing its derivative
$$f'(x)=-4*x^3+8*x$$, they find the critical points at $$x=-\sqrt{2}$$, 0, and $$x=\sqrt{2}$$. The researcher then mistakenly concludes that $$f(x)$$ is increasing on the interval $$(-\sqrt{2},\sqrt{2})$$ and decreasing elsewhere. Identify the mistake in this analysis.
There is no mistake; the function is indeed increasing on $$(-\sqrt{2}, \sqrt{2})$$.
The mistake is due to an incorrect factorization of the derivative; additional critical points were overlooked.
The error is in the sign analysis of the derivative; a proper test shows that the function is increasing on $$( -\infty, -\sqrt{2})$$ and $$(0, \sqrt{2})$$, and decreasing on $$(-\sqrt{2}, 0)$$ and $$(\sqrt{2}, \infty)$$.
The error is in omitting a constant term in the function, which alters its derivatives.