Q. 2 - Calculus

Question

More second-derivative graphs: The three graphs shown are graphs of a function $f$ and its first and second derivatives $f^{\text{'}}$ and width="18" style="max-width: none; vertical-align: -5px;" $f^{\text{'}\text{'}}$, in no particular order. Identify which graph is which.

Step-by-Step Solution

Answer

The graph of the function $f$ is represented by Graph I, its derivative by Graph II, and its second derivative by Graph III. 

Step 1: Given information

The graphs I, II, and III below show,

 

Step 2: Calculation

The point on the graph where the$f^{\text{'}}$ graph crosses the x-axis must be located in order to determine the critical values of the function.

The interval where the function decreases in the $f^{\text{'}}$ graph is where $f^{\text{'}}<0$ is below the x-axis.

The interval where the function increases in the $f^{\text{'}}$ graph is where $f^{\text{'}}>0$ is above the x-axis. The region where the function concaves down on the $f^{\text{'}\text{'}}$ graph is below the x-axis, where occurs.

The interval where the function convex in the $f^{\text{'}\text{'}}$graph is where $f^{\text{'}\text{'}}>0$ is above the x-axis. Use the critical number to write all the intervals when the function will be rising or falling.

The value of x where $f^{\text{'}}\left(x\right)=0$ is known as the critical number.

Determine if the crucial value is a local maximum or local minimum based on the situations below.

If the function is rising to the right and decreasing to the left of the critical number, then the local minimum at c is provided by $(c, f(c\left)\right)$.

There is a local maximum at c if the function to the left of the critical number is increasing and the function to the right is falling.

According to $(c, f(c\left)\right)$.

Graph:

Step 3: simplification

Interpretation:

There is a critical value for the function when the slope is equal to zero:

$f^{\text{'}}\left(x\right)=0$at $x=1$.

At $x=1$, the function has a local minimum.

The f graph shows a decrease in the $(-\infty ,1)$ range and an increase in the $(1,\infty )$ range.

The $f^{\text{'}}$graph is below the x-axis in the range of $\left(-\infty ,1\right)$ and above the x-axis in the range of .

The graph of $f^{\text{'}\text{'}}<0$ {below x - axis} and the function f is concave down in the interval $(-\infty ,0)$.

The function f''s graph is concave up in the range $\left(0,\infty \right)$ and is above the x-axis.