Part 3/6:
Categories of Critical Points
Local Minimum: If the derivative changes from negative to positive at a critical point, this indicates a local minimum.
Local Maximum: If the derivative shifts from positive to negative at a critical point, this indicates a local maximum.
Neither: If the derivative does not change signs (remains positive or remains negative), the critical point is neither a maximum nor a minimum.
Applying the First Derivative Test
To illustrate the application of the First Derivative Test, let us consider an example function:
Example Function
Let:
[ f(x) = 3x^4 - 4x^3 - 12x^2 + 5 ]
To find local maxima or minima, we begin by computing the derivative:
[ f'(x) = 12x^3 - 12x^2 - 24x ]