Intervals of constancy of sign are an important aspect of function analysis that help determine on which intervals a function maintains the same value (positive or negative). Here are some formulas and steps that can help in determining intervals of constancy of sign:
- Finding the derivative of a function:
- If ( f(x) ) is your function, then find its derivative ( f'(x) ).
- Definition of critical points:
- Find points where ( f'(x) = 0 ) or ( f'(x) ) does not exist. These points may be potential boundaries of intervals of sign constancy.
- Construction of the signed interval:
- Divide the number line into intervals using the critical points found.
- Select test points from each interval.
- Determining the sign of the derivative:
- Substitute the test points into the derivative ( f'(x) ) and determine the sign (positive or negative) on each interval.
- Intervals of sign constancy:
- If ( f'(x) > 0 ) on an interval, then the function increases on this interval.
- If ( f'(x) < 0 ) on an interval, then the function decreases on this interval.
- If ( f'(x) = 0 ) at a point, then check whether the sign of the derivative changes when passing through this point.
- Recording intervals:
- Write down the intervals where the function retains its sign (positive or negative).
Example:
- For the function ( f(x) = x^3 – 3x^2 + 4 ):
- Find the derivative: ( f'(x) = 3x^2 – 6x ).
- Find the critical points: (3x(x – 2) = 0) → (x = 0) and (x = 2).
- Conduct a sign analysis on the intervals ( (-\infty, 0) ), ( (0, 2) ), ( (2, +\infty) ).