Understanding Concavity and Points of Inflection

This video tutorial provides additional examples on the concavity of functions, focusing on transcendental functions. It reviews the concept of concavity, explaining how the second derivative is used to determine whether a function is concave up or down. The video includes two detailed examples: one using trigonometric functions and another using logarithmic functions. Each example demonstrates how to find the second derivative, determine test intervals, and identify points of inflection. The video concludes with a verification of results using graphs.

What is the primary test used to determine the concavity of a function?

• Second derivative test

• First derivative test

• Limit test

• Integral test

If the second derivative of a function is positive over an interval, what can be said about the graph?

• The graph is constant.

• The graph is linear.

• The graph is concave downward.

• The graph is concave upward.

What are possible points of inflection?

• Where the first derivative is zero

• Where the second derivative is zero or undefined

• Where the function is decreasing

• Where the function is increasing

In Example 1, what trigonometric function is used to find the second derivative?

• Tangent and Cotangent

• Secant and Cosecant

• Sine and Cosine

• Arcsine and Arccosine

What is the significance of the points where the second derivative changes sign?

• They are points of symmetry.

• They are points of discontinuity.

• They are points of inflection.

• They are points of maximum curvature.

In Example 2, which rule is applied to find the second derivative?

• Quotient rule

• Chain rule

• Product rule

• Power rule

What is the domain condition for the function in Example 2?

• X is equal to zero

• X is greater than zero

• X is not equal to zero

• X is less than zero

What is the approximate value of e to the power of -3/2 used in Example 2?

• 0.223

• 1.5

• 2.718

• 0.5

How is the point of inflection verified in the examples?

• By using a calculator

• By using a graph

• By using a table

• By using a ruler

What does a negative second derivative indicate about the function's concavity?

• The function is concave upward.

• The function is concave downward.

• The function is linear.

• The function is constant.