Question

Suppose $f$and role="math" localid="1648545106369" $f\text{'}$are differentiable on an interval role="math" localid="1648545118219" $I$. If ____ is positive on role="math" localid="1648545131434" $I$, then $f$is concave up on $I$. If ____ is on ____$I$, then $f$ is concave down on $I$.

Short answer

Suppose $f$and $f\text{'}$are differentiable on an interval $I$. If $f\text{'}$is positive on $I$, then $f$is concave up on $I$. If localid="1648551050926" $f\text{'}$is negative on $I$, then $f$is concave down on I.

Step-by-step solution

Step 1. Given information 

The given statement is,

Suppose $f$and $f\text{'}$are differentiable on an interval $I$. If ____ is positive on $I$, then $f$is concave up on $I$. If ____ is on ____ $I$, then $f$is concave down on localid="1648547797062" $I$.

Step 2. Fill in the blanks  

The given statement is a theorem

That is,

Suppose $f$and $f\text{'}$are differentiable on an interval $I$. If $f\text{'}$is positive on $I$, then $f$is concave up on $I$. If localid="1648551100145" $f\text{'}$is negative on $I$, then $f$is concave down on I.