Calculus I

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Concave up

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Calculus I

Definition

A function is concave up on an interval if its second derivative is positive over that interval. Graphically, this means the curve opens upwards like a cup.

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5 Must Know Facts For Your Next Test

  1. A function $f(x)$ is concave up on an interval if $f''(x) > 0$ for all $x$ in that interval.
  2. Concavity can change at points where the second derivative is zero or undefined, known as inflection points.
  3. If a function is concave up, its first derivative $f'(x)$ is increasing.
  4. The tangent line to the curve will lie below the graph of the function when it is concave up.
  5. Concavity can be tested using the second derivative test: if $f''(c) > 0$, then $f(c)$ has a local minimum at $c$.

Review Questions

  • How do you determine if a function is concave up on a given interval?
  • What does it mean for the first derivative of a function when the function is concave up?
  • Explain how you would use the second derivative test to find local minima.
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