Determine the Open Intervals on Which the Graph of the Function

question 69

Multiple Choice

Determine the open intervals on which the graph of the function
$f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 625 }$ is concave upward or concave downward.

• A) concave upward: \(\left( - \infty , - \frac { 25 \sqrt { 3 } } { 3 } \right) , \left( \frac { 25 \sqrt { 3 } } { 3 } , \infty \right) ;\) concave downward: \(\left( - \frac { 25 \sqrt { 3 } } { 3 } , \frac { 25 \sqrt { 3 } } { 3 } \right)\)
• B) concave upward: \(\left( - \frac { 25 \sqrt { 3 } } { 3 } , \frac { 25 \sqrt { 3 } } { 3 } \right) ;\) concave downward: \(\left( - \infty , - \frac { 25 \sqrt { 3 } } { 3 } \right) , \left( \frac { 25 \sqrt { 3 } } { 3 } , \infty \right)\)
• C) concave upward: \(( - \infty , 0 )\); concave downward: \(( 0 , \infty )\)
• D) concave upward: \(\left( - \frac { 625 } { 3 } , \frac { 625 } { 3 } \right) ;\) concave downward: \(\left( - \infty , - \frac { 625 } { 3 } \right) , \left( \frac { 625 } { 3 } , \infty \right)\)
• E) concave upward: \(\left( - \infty , - \frac { 625 } { 3 } \right) , \left( \frac { 625 } { 3 } , \infty \right) ;\) concave downward: \(\left( - \frac { 625 } { 3 } , \frac { 625 } { 3 } \right)\)

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Definitions:

Economic and Military Predictions

The process of forecasting future economic trends and military capabilities or situations based on current data and trends.

Correlational Design

A type of research methodology that examines the relationship between two or more variables without manipulating them.

Naturally Occurring Relationships

Relationships that form without any intervention or manipulation by researchers, reflecting genuine connections between entities.

Meta-Analysis

A statistical analysis that combines the results of multiple scientific studies to arrive at a comprehensive conclusion.