Calculus AP AB question: given the function f'(x)=-x(x-4)^3(x-2)(x+1) what conclusion can be drawn directly from the sign chart of the given function about the graph of f when x=2?

Question

Calculus AP AB question: given the function f'(x)=-x(x-4)^3(x-2)(x+1) what conclusion can be drawn directly from the sign chart of the given function about the graph of f when x=2?

Answer 1

Since the factored form of the derivative is given, it is trivial to find the critical points (they are x = -1, x = 0, x = 2, and x = 4).

We can try values of "x" on each side of x = 2 to create "signs" near x = 2.

At x = 1, the signs are -(+)(-)(-)(+) = - (meaning the function is decreasing)

At x = 3, the signs are -(+)(-)(+)(+) = + (meaning the function is increasing)

Therefore x = 2 is a local minimum.