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Derivatives and Shapes of Graphs- DA4

Objectives

  • Explain how the sign of the first derivative affects the shape of a function’s graph.
  • State the first derivative test for critical points.
  • Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.
  • Explain the concavity test for a function over an open interval.
  • Explain the relationship between a function and its first and second derivatives.
  • State the second derivative test for local extrema.

Summary

The critical numbers of a continuous function f are the values of p for which f ′ ( p ) = 0 or f ′ ( p ) does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.

Given a differentiable function f , whenever f ′ is positive, f is increasing; whenever f ′ is negative, f is decreasing. The first derivative test tells us that at any point where f changes from increasing to decreasing, f has a local maximum, while conversely at any point where f changes from decreasing to increasing f has a local minimum.

Given a twice differentiable function f , if we have a horizontal tangent line at x = p and f ″ ( p ) is nonzero, the sign of f ″ tells us the concavity of f and hence whether f has a maximum or minimum at x = p . In particular, if f ′ ( p ) = 0 and f ″ ( p ) < 0 , then f is concave down at p and f has a local maximum there, while if f ′ ( p ) = 0 and f ″ ( p ) > 0 , then f has a local minimum at p . If f ′ ( p ) = 0 and f ″ ( p ) = 0 , then the second derivative does not tell us whether f has a local extreme at p or not.

 

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