Explanation Given information: The horizontal line y = c intersects the function f ( x ) = x 2 + r x + s at two points ( x 1 , f ( x 1 ) ) and ( x 2 , f ( x 2 ) ) is redrawn as shown in Figure 1. Concept used: If the second derivative of the function f ( x ) is positive then the function f ( x ) has minimum value and if the second derivative of the function f ( x ) is negative then the function has maximum value. Proof: Figure 1 represents the graph of parabola f ( x ) = x 2 + r x + s . To obtain the critical point of the function f ( x ) , differentiate the function f ( x ) as shown below: d d x f ( x ) = d d x ( x 2 + r x + s ) f ′ ( x ) = 2 x + r To obtain the critical point of the function, equate the function f ′ ( x ) to 0 . f ′ ( x ) = 0 2 x + r = 0 x = − r 2 The critical point is x = − r 2 . Now, differentiate the function f ′ ( x ) with respect to x as shown below: d d x f ′ ( x ) = d d x ( 2 x + r ) f ″ ( x ) = 2 Since, the value of f ″ ( x ) is positive so, the function f ( x ) has a minima at x = − r 2 . From Figure 1 it is observed that the horizontal line y = c intersects the graph of the function f ( x ) = x 2 + r x + s at point ( x 1 , c ) and ( x 2 , c ) . Now, substitute x 1 for x and c for f ( x ) in the function f ( x ) = x 2 + r x + s as shown below: c = x 1 2 + r x 1 + s

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 4.2, Problem 98E

To determine

To show:

That the minimum value of the function f(x)=x2+rx+s takes at midpoint M=x1+x22, if the horizontal line intersects the graph of function f(x) at two points.

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Chapter 4.2, Problem 97E

Chapter 4.2, Problem 99E

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Chapter 4 Solutions

Calculus: Early Transcendentals

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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That the minimum value of the function f ( x ) = x 2 + r x + s takes at midpoint M = x 1 + x 2 2 , if the horizontal line intersects the graph of function f ( x ) at two points.

Solution Summary: The author explains that the minimum value of the function f(x) = 2+rx+s takes at midpoint.

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That the minimum value of the function f(x)=x2+rx+s takes at midpoint M=x1+x22, if the horizontal line intersects the graph of function f(x) at two points.

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Chapter 4 Solutions

That the minimum value of the function f ( x ) = x 2 + r x + s takes at midpoint M = x 1 + x 2 2 , if the horizontal line intersects the graph of function f ( x ) at two points.

Solution Summary: The author explains that the minimum value of the function f(x) = 2+rx+s takes at midpoint.

Concept explainers

Videos

To show: That the minimum value of the function f(x)=x2+rx+s takes at midpoint M=x1+x22, if the horizontal line intersects the graph of function f(x) at two points.

Want to see the full answer?

Chapter 4 Solutions

To show:

That the minimum value of the function f(x)=x2+rx+s takes at midpoint M=x1+x22, if the horizontal line intersects the graph of function f(x) at two points.