Explanation Given: F ( x ) = ∫ 0 x cos ( t 3 / 2 ) d t Key concepts applied: FTC II or The second Fundamental Theorem of Calculus deals with derivatives of definite integrals with respect to variable upper limits of integration. It states that if F ( x ) = ∫ a x f ( t ) d t and f is continuous, then F ' ( x ) = f ( x ) . Critical points are those points at which the derivative of f ( x ) with respect to x is equal to zero corresponds to a maximum, a minimum or an inflexion point of the function f(x). The basic properties of the maxima and minima are summarized in the following table Behavior Graph Derivative at point slightly to the left of the extremum point x o Derivative at extremum point x o Derivative at point slightly to the right of the extremum point x o Second derivative at extremum point x o Local maximum f ' ( x o − ) > 0 Positive Increasing f ' ( x o ) = 0 Zero f ' ( x o + ) < 0 Negative Decreasing f ' ' ( x o ) < 0 Local Minimum f ' ( x o − ) < 0 Negative Decreasing f ' ( x o ) = 0 Zero f ' ( x o + ) > 0 Positive Increasing f ' ' ( x o ) > 0 From the above table we understand that the minimum values of a function occur where its derivate goes from negative to positive, similarly the maximum values of a function occur where its derivate goes from positive to negative. Correspondingly, the minimum values of a function occur where its second derivate is negative, and the maximum values of a function occur where its second derivate is positive. Inflection point Some functions have an inflexion point which is a point at which a change in the direction of curvature occurs. At inflexion points, the second derivative is equal to zero. Concavity of a graph A graph is said to be concave up at a point if the tangent line to the graph at that point lies below the graph in the vicinity of the point Regardless of the sign of the slope (positive, negative or zero), the slopes of the tangent are increasing (from left to right) when it is concave up. Therefore, when a function is concave up, the graph of its derivative is increasing. A graph is said to be concave down at a point if the tangent line to the graph at that point lies above the graph in the vicinity of the point. Regardless of the sign of the slope (positive, negative or zero), the slopes of the tangent are decreasing as we move from left to right when the graph is concave down. Therefore, when a function is concave down, the graph of its derivative is decreasing. A point where the concavity changes (from up to down or down to up) is called a point of inflection (POI); note that the tangent line to a graph at a point of inflection must cross the graph at that point. Calculation: Description F ( x ) = ∫ 0 x cos ( t 3 / 2 ) d t Step 1...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 5.5, Problem 49E

To determine

The smallest positive critical point of the given function and to determine if it is a local min or max

The smallest inflection point of the given function and how the concavity changes using graph

Expert Solution & Answer

Chapter 5 Solutions

Calculus - Standalone book

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

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The smallest positive critical point of the given function and to determine if it is a local min or max The smallest inflection point of the given function and how the concavity changes using graph

Solution Summary: The author explains the fundamental theorem of calculus, which deals with derivatives of definite integrals with respect to variable upper limits of integration.

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The smallest positive critical point of the given function and to determine if it is a local min or max The smallest inflection point of the given function and how the concavity changes using graph

Chapter 5 Solutions

The smallest positive critical point of the given function and to determine if it is a local min or max

The smallest inflection point of the given function and how the concavity changes using graph