F''(c) > 0

Definition

The expression f''(c) > 0 indicates that the second derivative of a function f at a point c is positive, which suggests that the function is concave up at that point. This implies that the graph of the function is bending upwards, and if f'(c) = 0, it confirms that c is a local minimum. Understanding this concept is essential when applying the second derivative test to find local extrema in functions.

5 Must Know Facts For Your Next Test

1. If f''(c) > 0, it indicates that the slope of f' is increasing at point c, meaning f' changes from negative to positive.
2. When using the second derivative test, if f''(c) > 0 and f'(c) = 0, then c is a local minimum.
3. Graphically, when f''(c) > 0, the tangent line at point c lies below the graph of f in a neighborhood around c.
4. This condition helps to identify points of inflection where the concavity of the function changes.
5. The second derivative test is often preferred over the first derivative test for its simplicity in determining local extrema.

Review Questions

• How can you interpret f''(c) > 0 in relation to the shape of the graph of a function?
  • When f''(c) > 0, it indicates that the function's graph is concave up at point c. This means that as you move away from c in either direction, the values of the function are increasing, creating a 'bowl' shape. This condition also implies that if there’s a critical point where f'(c) = 0, then c is likely a local minimum since the slope transitions from decreasing to increasing.
• Why is it important to consider both f'(c) and f''(c) when using the second derivative test for local extrema?
  • Using both f'(c) and f''(c) provides a more complete picture when identifying local extrema. If f'(c) = 0 indicates a potential extremum but does not specify whether it’s a maximum or minimum. By checking f''(c), we determine the concavity at that point. If f''(c) > 0, it confirms that we have a local minimum; conversely, if f''(c) < 0, we have a local maximum. This combination helps ensure accurate classification of critical points.
• Evaluate how changing conditions in a function might impact its second derivative and consequently affect the identification of local minima or maxima.
  • Changes in a function's behavior can directly influence its second derivative. For instance, if parameters within the function are altered, leading to new turning points or changing concavity, this will affect whether f''(c) remains positive or negative at those points. Consequently, if new critical points emerge where f' still equals zero but with varying signs in f'', it may lead to misidentification of local minima or maxima. Understanding these relationships allows for better predictions about how changes in one aspect of a function affect overall behavior.

Related terms

Concavity: The direction in which a curve bends, with concave up indicating the curve opens upward and concave down indicating it opens downward.

Local Minimum: A point on the graph of a function where the function value is lower than that of nearby points, often found where the first derivative equals zero and the second derivative is positive.

First Derivative Test: A method for finding local extrema by analyzing the sign changes of the first derivative before and after critical points.