Part 2/6:
To understand the First Derivative Test, it's essential to grasp the fundamentals of derivatives. The derivative, denoted as ( f'(x) ), effectively acts as the slope of a tangent line to the graph of the function ( f(x) ) at any given point. When the derivative is positive (( f'(x) > 0 )), the graph of the function is increasing. Conversely, if the derivative is negative (( f'(x) < 0 )), the function is decreasing.
Identifying Critical Points
Critical points are where the function’s derivative is either zero or undefined. These points are vital because they often serve as candidates for local maxima or minima. To apply the First Derivative Test, one must first find these critical points, denoted as ( c ), where ( f'(c) = 0 ) or where ( f' ) does not exist.