concave

Examples of concave in the following topics:

• Concavity and the Second Derivative Test

  • A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point.
  • Specifically, a twice-differentiable function $f$ is concave-up if $f''(x)$ is positive and concave-down if $f''(x)$ is negative.
  • If it is concave-up at the point, it is a minimum; if concave-down, it is a maximum.

• Derivatives and the Shape of the Graph

  • The shape of a graph may be found by taking derivatives to tell you the slope and concavity.
  • At an inflection point, a function switches from being a convex function to being a concave function or vice versa.
  • Sketch the shape of a graph by using differentiation to find the slope and concavity

• Linear Approximation

  • If $f$ is concave-down in the interval between $x$ and $a$, the approximation will be an overestimate (since the derivative is decreasing in that interval).
  • If $f$ is concave-up, the approximation will be an underestimate.