For \(f(x)=x^2-4x+3\), find the vertex and state where the graph is increasing and decreasing.

For \(f(x)=x^3-3x^2\), find the critical points and classify each as a local maximum, local minimum, or neither.

For \(f(x)=x^3-6x\), find the intervals where the graph is concave up and concave down.

For \(f(x)=x^4-4x^2\), find the critical points and determine where the function is increasing.

For \(f(x)=\frac{x+1}{x-2}\), find the vertical and horizontal asymptotes.

For \(f(x)=x^3-3x+1\), find the local extrema and the inflection point.

For \(f(x)=\frac{1}{x^2+1}\), determine where the function is increasing and decreasing.

For \(f(x)=x^4\), explain whether \(x=0\) is an inflection point.

For \(f(x)=x+\frac{4}{x}\), with domain \(x\ne0\), find the critical points and classify them.

For \(f(x)=\frac{x^2}{x-1}\), find the domain, vertical asymptote, and critical points.