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Q. 60

Expert-verified
Found in: Page 276

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

The possible graph of f is

See the step by step solution

## 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 interval $\left(-\infty ,0\right)$ and negative on the interval $\left(0,\infty \right).$Thus, f will we increase on the positve intervals and decrease on the negative intervals.

$f\text{'}\text{'}$ is positive on the intervals $\left(-\infty ,-1\right)\mathrm{and}\left(1,\infty \right)$ and negative on the interval $\left(-1,1\right).$ Thus, f will be concave up on the positive intervals and concave down on the negative intervals.

The graph has roots at $x=-1,$ a local minimum at $x=0,$ and inflection points at $x=-1and1.$

The graph is