Q. 55 - Calculus

Question

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

Step-by-Step Solution

Answer

The possible graph of f  is 

Step 1. Given Information.

The given sign chart is 

Step 2. Sketch the graph of f.

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, then f is concave up on I.

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

Step 3. The graph of f.

From the given chart, we conclude that

$f\text{'}$ is  positive on the intervals $\left(-\infty ,-2\right) \mathrm{and} \left(-2,\infty \right).$Thus, f will we increase on the intervals.

$f\text{'}\text{'}$ is positive on the interval $\left(-2,\infty \right)$ and negative on the interval $\left(-\infty ,-2\right).$ Thus, f will be concave up on the positive intervals and concave down on the negative intervals.

The graph is