Practice with the First Derivative Test

Consider the function $$f(x) = x^2 - 4x + 3$$ . Find the critical points and determine if they are local maxima, minima, or neither using the First Derivative Test.

For the function $$h(x) = x^2 - 2x$$ , use Rolle's Theorem to determine if there is a point $$c$$ in the interval $$[0, 2]$$ where $$h'(c) = 0$$ .

Consider the function $$p(x) = x^2 - 6x + 9$$ . Determine if the function has any local extrema and classify them.

Given the function $$r(x) = x^2 - x$$ , determine the intervals where the function is decreasing.

Consider the function $$u(x) = x^3 - 3x + 2$$ . Use the First Derivative Test to determine the nature of the critical points.