Sketching the Graph of a Continuous Function with Derivatives

Math Problem Statement

The function f(x) is continuous on

left parenthesis negative infinity comma infinity right parenthesis(−∞,∞).

Use the given information to sketch the graph of f.

f(negative 6−6)equals=00,

f(00)equals=negative 12−12,

f(66)equals=00;

f prime left parenthesis 0 right parenthesis equals 0f′(0)=0,

f prime left parenthesis negative 6 right parenthesisf′(−6)

and

f prime left parenthesis 6 right parenthesisf′(6)

are not defined;

f prime left parenthesis x right parenthesisf′(x)greater than>0

left parenthesis 0 comma 6 right parenthesis(0,6)

and

left parenthesis 6 comma infinity right parenthesis(6,∞);

f prime left parenthesis x right parenthesisf′(x)less than<0

left parenthesis negative infinity comma negative 6 right parenthesis(−∞,−6)

and

left parenthesis negative 6 comma 0 right parenthesis(−6,0);

f double prime left parenthesis negative 6 right parenthesisf′′(−6)

and

f double prime left parenthesis 6 right parenthesisf′′(6)

are not defined;

f double prime left parenthesis x right parenthesisf′′(x)greater than>0

left parenthesis negative 6 comma 6 right parenthesis(−6,6);

f double prime left parenthesis x right parenthesisf′′(x)less than<0

left parenthesis negative infinity comma negative 6 right parenthesis(−∞,−6)

and left parenthesis 6 comma infinity right parenthesis(6,∞)

Question content area bottom

Part 1

Choose the correct graph of f below

-1414-1414xy

A coordinate system has a horizontal x-axis labeled from negative 14 to 14 in increments of 2 and a vertical y-axis labeled from negative 14 to 14 in increments of 2. From left to right, a curve falls at an increasing rate to (negative 6, 0), rises at a decreasing rate to a maximum at (0, 12), falls at an increasing rate to (6, 0), and then rises at a decreasing rate.

-1414-1414xy

A coordinate system has a horizontal x-axis labeled from negative 14 to 14 in increments of 2 and a vertical y-axis labeled from negative 14 to 14 in increments of 2. From left to right, a curve rises at a constant rate to (negative 6, 0), falls at a decreasing rate to a minimum at (0, negative 12), rises at an increasing rate to (6, 0), and then falls at a constant rate.

-1414-1414xy

A coordinate system has a horizontal x-axis labeled from negative 14 to 14 in increments of 2 and a vertical y-axis labeled from negative 14 to 14 in increments of 2. From left to right, a curve falls at an increasing rate to (negative 6, 0), falls at a decreasing rate to a minimum at (0, negative 12), rises at an increasing rate to (6, 0), and then rises at a decreasing rate.

-1414-1414xy

Solution

Given the properties of the function $f(x)$ , let's analyze each piece of information to deduce the correct graph:

Points on the Graph:
- $f(-6) = 0$ : The function passes through $(-6, 0)$ .
- $f(0) = -12$ : The function passes through $(0, -12)$ .
- $f(6) = 0$ : The function passes through $(6, 0)$ .
Derivative Information:
- $f'(0) = 0$ : There's a horizontal tangent at $x = 0$ , indicating a local minimum or maximum at $(0, -12)$ .
- $f'(-6)$ and $f'(6)$ are undefined: These points likely correspond to sharp turns or cusps at $(-6, 0)$ and $(6, 0)$ .
- $f'(x) > 0$ on $(0, 6)$ and $(6, \infty)$ : The function is increasing on these intervals.
- $f'(x) < 0$ on $(-\infty, -6)$ and $(-6, 0)$ : The function is decreasing on these intervals.
Concavity Information:
- $f''(-6)$ and $f''(6)$ are undefined: This suggests points of inflection at $(-6, 0)$ and $(6, 0)$ .
- $f''(x) > 0$ on $(-6, 6)$ : The function is concave up between $-6$ and $6$ .
- $f''(x) < 0$ on $(-\infty, -6)$ and $(6, \infty)$ : The function is concave down outside these intervals.
Shape of the Graph:
- Given the concavity and the derivative information, the function has a concave-down decreasing behavior as it approaches $(-6, 0)$ from the left, changes to concave-up decreasing from $(-6, 0)$ to $(0, -12)$ , then continues to concave-up increasing from $(0, -12)$ to $(6, 0)$ , and finally switches to concave-down increasing after $(6, 0)$ .

Conclusion:

Based on these characteristics, the correct choice is:

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Math Problem Analysis

Mathematical Concepts

Calculus
Graphing Functions
Derivatives
Concavity

Formulas

f'(x) > 0 implies f is increasing
f'(x) < 0 implies f is decreasing
f''(x) > 0 implies f is concave up
f''(x) < 0 implies f is concave down

Theorems

Mean Value Theorem
First Derivative Test
Second Derivative Test

Suitable Grade Level

Grades 11-12

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