Concavity

Table Of Contents

Concavity

The polarity of the second interval.

Concave

• $f^{\prime \prime }(x)>0$
• All secant lines on the graph are above the graph
• Slopes increasing from left to right

Convex

• $f^{\prime \prime }(x)<0$
• All secant lines on the graph are below the graph
• Slopes decreasing from left to right

Concavity Test

Suppose both f and $f^{\prime }$ are differentiable on interval I.

• If $f^{\prime \prime }$ is positive on I, then $f$ is concave up
• If $f^{\prime \prime }$ is negative on I, then $f$ is concave down

Proof

$f^{\prime \prime }$ positive $⟹$ $f^{\prime }$ increasing by (inc/dec test) $⟺$ f concave up