For each set of sign charts in Exercises 53–62, sketch a possible

graph of f.

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" $\left(-\infty ,-1\right)and\left(2,\infty \right)$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 $\left(-\infty ,-1\right)and\left(2,\infty \right).$

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

The graph is