Identifying Function Monotonicity Using Derivative Graphs

To identify the monotonicity of a function based on the graph of its derivative, we need to understand the relationship between the derivative and the function’s monotonicity. The derivative describes the instantaneous rate of change of the function at a given point. If the derivative is positive, the function is increasing at that point; if the derivative is negative, the function is decreasing at that point; if the derivative is zero, the function may have a local extremum or an inflection point at that point.

The specific steps are as follows:

1. Observe the Value of the Derivative:

• The sign of the derivative f′(x)f'(x)f′(x) determines the monotonicity of the original function f(x)f(x)f(x):
• When f′(x)>0f'(x) > 0f′(x)>0, the function f(x)f(x)f(x) is increasing in that interval.
• When f′(x)<0f'(x) < 0f′(x)<0, the function f(x)f(x)f(x) is decreasing in that interval.
• When f′(x)=0f'(x) = 0f′(x)=0, it may be a point of extremum or an inflection point.

2. Identify the Zeros of the Derivative:

• The zeros of the derivative are the stationary points of the original function (i.e., possible extremum points or flat points).
• Find the intersections of the derivative graph with the xxx-axis, i.e., the positions where f′(x)=0f'(x) = 0f′(x)=0.
• At these zeros, the original function may change from increasing to decreasing or from decreasing to increasing.

3. Analyze the Sign Changes of the Derivative:

• On either side of each zero of the derivative, check the sign of f′(x)f'(x)f′(x):
• If f′(x)f'(x)f′(x) changes from positive to negative, then the original function changes from increasing to decreasing at that point (maximum point).
• If f′(x)f'(x)f′(x) changes from negative to positive, then the original function changes from decreasing to increasing at that point (minimum point).
• If f′(x)f'(x)f′(x) does not change sign, then the original function may have a flat point or an inflection point at that point.

4. Combine Derivative Graphs to Analyze Monotonicity:

• Based on the positive and negative regions of the derivative graph, divide the xxx-axis into several intervals.
• If the sign of the derivative is consistent within each interval, then the original function is monotonic in that interval:
• f′(x)>0
  
  f'(x) > 0f′(x)>0

  : The original function is increasing.
• f′(x)<0
  
  f'(x) < 0f′(x)<0

  : The original function is decreasing.

Example Analysis:

Assume the graph of the derivative f′(x)f'(x)f′(x) is as follows:

• On the interval (−∞,−2)(-
  \infty, -2)(−∞,−2), f′(x)>0f'(x) > 0f′(x)>0, the original function is increasing.
• On the interval (−2,1)(-2, 1)(−2,1), f′(x)<0f'(x) < 0f′(x)<0, the original function is decreasing.
• On the interval (1,∞)(1,
  \infty)(1,∞), f′(x)>0f'(x) > 0f′(x)>0, the original function is increasing.

Conclusion:

• f(x)
  
  f(x)f(x)

  has a maximum value at x=−2x = -2x=−2 and a minimum value at x=1x = 1x=1.
• The monotonicity is as follows:
• (−∞,−2):f(x)
  
  (-
  \infty, -2): f(x)(−∞,−2):f(x)

  is increasing;
• (−2,1):f(x)
  
  (-2, 1): f(x)(−2,1):f(x)

  is decreasing;
• (1,∞):f(x)
  
  (1,
  \infty): f(x)(1,∞):f(x)

  is increasing.