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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A function $f(x)$ is concave up on an interval if $f''(x) > 0$ for all $x$ in that interval.
Concavity can change at points where the second derivative is zero or undefined, known as inflection points.
If a function is concave up, its first derivative $f'(x)$ is increasing.
The tangent line to the curve will lie below the graph of the function when it is concave up.
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.
Related terms
Inflection Point: A point on the graph where the concavity changes from concave up to concave down or vice versa.
Second Derivative Test: A test used to determine local maxima and minima by analyzing the sign of the second derivative at critical points.
Concave Down: A function is concave down on an interval if its second derivative is negative over that interval, making the curve open downwards like an arch.