91Ó°ÊÓ

The given sign chart is

To sketch the possible graph of f, we will use theorem 3.6 and 3.10.

Theorem 3.6 states that the Derivative Measures Where a Function is Increasing or Decreasing, let f be a function that is differentiable on an interval I.

(a) If $f\text{'}$ is positive in the interior of I, then f is increasing on I.

(b) If $f\text{'}$is negative in the interior of I, then f is decreasing on I.

(c) If $f\text{'}$ is zero in the interior of I, then f is constant on I.

Theorem 3.10 states that the Second Derivative Determines Concavity, suppose both f and $f\text{'}$ are differentiable on an interval I.

(a) If $f\text{'}\text{'}$ is positive on I, thenf is concave up on I.

(b) If $f\text{'}\text{'}$is negative onI, then f is concave down on I.

From the given chart, we conclude that

$f\text{'}$is negative on the intervals role="math" localid="1648452442973" $\left(-∞,-1\right)and\left(2,∞\right)$and positive on the intervals role="math" localid="1648452435152" $\left(-1,2\right).$ Thus, f will we increase on the interval $(-1,2)$and decrease on the intervals $\left(-∞,-1\right)and\left(2,∞\right).$

$f\text{'}\text{'}$is positive on the interval $\left(-∞,0.5\right)$and negative on the interval $\left(0.5,∞\right)$. Thus, fwill be concave up on the positive intervals and concave down on the negative intervals.

The graph is